Theorem eleq2dALT

Description: Alternate proof of eleq2d , shorter at the expense of requiring ax-12 . (Contributed by NM, 27-Dec-1993) (Revised by Wolf Lammen, 20-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	eleq1d.1	|- ( ph -> A = B )
Assertion	eleq2dALT	|- ( ph -> ( C e. A <-> C e. B ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1d.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	anbi2d	|- ( ph -> ( ( x = C /\ x e. A ) <-> ( x = C /\ x e. B ) ) )
6	5	exbidv	|- ( ph -> ( E. x ( x = C /\ x e. A ) <-> E. x ( x = C /\ x e. B ) ) )
7		dfclel	|- ( C e. A <-> E. x ( x = C /\ x e. A ) )
8		dfclel	|- ( C e. B <-> E. x ( x = C /\ x e. B ) )
9	6 7 8	3bitr4g	|- ( ph -> ( C e. A <-> C e. B ) )