Theorem eleq2w 2258

Description: Weaker version of eleq2 2260 (but more general than elequ2 2172) not depending on ax-ext 2178 nor df-cleq 2189. (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 2172	. . . 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 1839	. 2 |- ( x = y -> ( E. z ( z = A /\ z e. x ) <-> E. z ( z = A /\ z e. y ) ) )
4		df-clel 2192	. 2 |- ( A e. x <-> E. z ( z = A /\ z e. x ) )
5		df-clel 2192	. 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 1364   E.wex 1506   e. wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-14 2170

This theorem depends on definitions:  df-bi 117  df-clel 2192

This theorem is referenced by:  exmidontriimlem4  7291