Theorem eqeq1dALT

Description: Alternate proof of eqeq1d , shorter but requiring ax-12 . (Contributed by NM, 27-Dec-1993) (Revised by Wolf Lammen, 19-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	eqeq1d.1	|- ( ph -> A = B )
Assertion	eqeq1dALT	|- ( ph -> ( A = C <-> B = C ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1d.1	|- ( ph -> A = B )
2		dfcleq	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
3	1 2	sylib	|- ( ph -> A. x ( x e. A <-> x e. B ) )
4	3	19.21bi	|- ( ph -> ( x e. A <-> x e. B ) )
5	4	bibi1d	|- ( ph -> ( ( x e. A <-> x e. C ) <-> ( x e. B <-> x e. C ) ) )
6	5	albidv	|- ( ph -> ( A. x ( x e. A <-> x e. C ) <-> A. x ( x e. B <-> x e. C ) ) )
7		dfcleq	|- ( A = C <-> A. x ( x e. A <-> x e. C ) )
8		dfcleq	|- ( B = C <-> A. x ( x e. B <-> x e. C ) )
9	6 7 8	3bitr4g	|- ( ph -> ( A = C <-> B = C ) )