Theorem eleq2w 2149

Description: Weaker version of eleq2 2151 (but more general than elequ2 1648) not depending on ax-ext 2070 nor df-cleq 2081. (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 1648	. . . 4 |- ( x = y -> ( z e. x <-> z e. y ) )
2	1	anbi2d 452	. . 3 |- ( x = y -> ( ( z = A /\ z e. x ) <-> ( z = A /\ z e. y ) ) )
3	2	exbidv 1753	. 2 |- ( x = y -> ( E. z ( z = A /\ z e. x ) <-> E. z ( z = A /\ z e. y ) ) )
4		df-clel 2084	. 2 |- ( A e. x <-> E. z ( z = A /\ z e. x ) )
5		df-clel 2084	. 2 |- ( A e. y <-> E. z ( z = A /\ z e. y ) )
6	3, 4, 5	3bitr4g 221	1 |- ( x = y -> ( A e. x <-> A e. y ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   E.wex 1426   e. wcel 1438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-clel 2084

This theorem is referenced by: (None)