Theorem eleq2w 2232

Description: Weaker version of eleq2 2234 (but more general than elequ2 2146) not depending on ax-ext 2152 nor df-cleq 2163. (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 2146	. . . 4 |- ( x = y -> ( z e. x <-> z e. y ) )
2	1	anbi2d 461	. . 3 |- ( x = y -> ( ( z = A /\ z e. x ) <-> ( z = A /\ z e. y ) ) )
3	2	exbidv 1818	. 2 |- ( x = y -> ( E. z ( z = A /\ z e. x ) <-> E. z ( z = A /\ z e. y ) ) )
4		df-clel 2166	. 2 |- ( A e. x <-> E. z ( z = A /\ z e. x ) )
5		df-clel 2166	. 2 |- ( A e. y <-> E. z ( z = A /\ z e. y ) )
6	3, 4, 5	3bitr4g 222	1 |- ( x = y -> ( A e. x <-> A e. y ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-14 2144

This theorem depends on definitions:  df-bi 116  df-clel 2166

This theorem is referenced by:  exmidontriimlem4  7201