Theorem eleq2w 2267

Description: Weaker version of eleq2 2269 (but more general than elequ2 2181) not depending on ax-ext 2187 nor df-cleq 2198. (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 2181	. . . 4 |- ( x = y -> ( z e. x <-> z e. y ) )
2	1	anbi2d 464	. . 3 |- ( x = y -> ( ( z = A /\ z e. x ) <-> ( z = A /\ z e. y ) ) )
3	2	exbidv 1848	. 2 |- ( x = y -> ( E. z ( z = A /\ z e. x ) <-> E. z ( z = A /\ z e. y ) ) )
4		df-clel 2201	. 2 |- ( A e. x <-> E. z ( z = A /\ z e. x ) )
5		df-clel 2201	. 2 |- ( A e. y <-> E. z ( z = A /\ z e. y ) )
6	3, 4, 5	3bitr4g 223	1 |- ( x = y -> ( A e. x <-> A e. y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1515   e. wcel 2176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-14 2179

This theorem depends on definitions:  df-bi 117  df-clel 2201

This theorem is referenced by:  exmidontriimlem4  7338