Theorem rspceeqv 2777

Description: Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)

Hypothesis

Ref	Expression
rspceeqv.1	|- ( x = A -> C = D )

Assertion

Ref	Expression
rspceeqv	|- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Distinct variable groups:   x, A   x, B   x, D   x, E

Allowed substitution hint:   C( x)

Proof of Theorem rspceeqv

Step	Hyp	Ref	Expression
1		rspceeqv.1	. . 3 |- ( x = A -> C = D )
2	1	eqeq2d 2126	. 2 |- ( x = A -> ( E = C <-> E = D ) )
3	2	rspcev 2760	1 |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1314   e. wcel 1463   E.wrex 2391

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659

This theorem is referenced by:  elixpsn  6583  ixpsnf1o  6584  elfir  6813  0ct  6944  ctmlemr  6945  ctssdclemn0  6947  fodju0  6969  mertenslemi1  11196  mertenslem2  11197  ctiunctlemfo  11795  elrestr  11971  restopnb  12193  mopnex  12494  metrest  12495