Theorem eqseq ≪ | index | src | ≫

theorem eqseq (_A1 _A2 _B1 _B2: set):
  $ _A1 == _A2 -> _B1 == _B2 -> (_A1 == _B1 <-> _A2 == _B2) $;

Step	Hyp	Ref	Expression
1		anl	_A1 == _A2 /\ _B1 == _B2 -> _A1 == _A2
2		anr	_A1 == _A2 /\ _B1 == _B2 -> _B1 == _B2
3	1, 2	eqseqd	_A1 == _A2 /\ _B1 == _B2 -> (_A1 == _B1 <-> _A2 == _B2)
4	3	exp	_A1 == _A2 -> _B1 == _B2 -> (_A1 == _B1 <-> _A2 == _B2)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_12), axs_set (ax_8)