Theorem eleq2w 2685

Description: Weaker version of eleq2 2690 (but more general than elequ2 2004) not depending on ax-ext 2602 (nor ax-12 2047 nor df-cleq 2615). (Contributed by BJ, 29-Sep-2019.)

Assertion

Ref	Expression
eleq2w	|- ( x = y -> ( A e. x <-> A e. y ) )

Proof of Theorem eleq2w

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2004	. . . 4 |- ( x = y -> ( z e. x <-> z e. y ) )
2	1	anbi2d 740	. . 3 |- ( x = y -> ( ( z = A /\ z e. x ) <-> ( z = A /\ z e. y ) ) )
3	2	exbidv 1850	. 2 |- ( x = y -> ( E. z ( z = A /\ z e. x ) <-> E. z ( z = A /\ z e. y ) ) )
4		df-clel 2618	. 2 |- ( A e. x <-> E. z ( z = A /\ z e. x ) )
5		df-clel 2618	. 2 |- ( A e. y <-> E. z ( z = A /\ z e. y ) )
6	3, 4, 5	3bitr4g 303	1 |- ( x = y -> ( A e. x <-> A e. y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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

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

This theorem is referenced by:  usgredgleordALT  26126  vtxdushgrfvedglem  26385  lmbr3  39979