Theorem eleq2dALT 2688

Description: Alternate proof of eleq2d 2687, shorter at the expense of using more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eleq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
eleq2dALT	|- ( ph -> ( C e. A <-> C e. B ) )

Proof of Theorem eleq2dALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1d.1	. . . . . 6 |- ( ph -> A = B )
2		dfcleq 2616	. . . . . 6 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
3	1, 2	sylib 208	. . . . 5 |- ( ph -> A. x ( x e. A <-> x e. B ) )
4	3	19.21bi 2059	. . . 4 |- ( ph -> ( x e. A <-> x e. B ) )
5	4	anbi2d 740	. . 3 |- ( ph -> ( ( x = C /\ x e. A ) <-> ( x = C /\ x e. B ) ) )
6	5	exbidv 1850	. 2 |- ( ph -> ( E. x ( x = C /\ x e. A ) <-> E. x ( x = C /\ x e. B ) ) )
7		df-clel 2618	. 2 |- ( C e. A <-> E. x ( x = C /\ x e. A ) )
8		df-clel 2618	. 2 |- ( C e. B <-> E. x ( x = C /\ x e. B ) )
9	6, 7, 8	3bitr4g 303	1 |- ( ph -> ( C e. A <-> C e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by: (None)