theorem sbeqd (_G: wff) {x: nat} (_a1 _a2: nat x) (_p1 _p2: wff x): $ _G -> _a1 = _a2 $ > $ _G -> (_p1 <-> _p2) $ > $ _G -> ([_a1 / x] _p1 <-> [_a2 / x] _p2) $;
Step | Hyp | Ref | Expression |
---|---|---|---|
1 |
_G -> y = y |
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2 |
hyp _ah |
_G -> _a1 = _a2 |
|
3 |
_G -> (y = _a1 <-> y = _a2) |
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4 |
_G -> (x = y <-> x = y) |
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5 |
hyp _ph |
_G -> (_p1 <-> _p2) |
|
6 |
_G -> (x = y -> _p1 <-> x = y -> _p2) |
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7 |
_G -> (A. x (x = y -> _p1) <-> A. x (x = y -> _p2)) |
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8 |
_G -> (y = _a1 -> A. x (x = y -> _p1) <-> y = _a2 -> A. x (x = y -> _p2)) |
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9 |
_G -> (A. y (y = _a1 -> A. x (x = y -> _p1)) <-> A. y (y = _a2 -> A. x (x = y -> _p2))) |
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10 |
conv sb |
_G -> ([_a1 / x] _p1 <-> [_a2 / x] _p2) |