Theorem rspceeqv 2925

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 2241	. 2 |- ( x = A -> ( E = C <-> E = D ) )
3	2	rspcev 2907	1 |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801

This theorem is referenced by:  elixpsn  6882  ixpsnf1o  6883  elfir  7140  0ct  7274  ctmlemr  7275  ctssdclemn0  7277  fodju0  7314  ccats1pfxeqrex  11247  mertenslemi1  12046  mertenslem2  12047  nninfctlemfo  12561  pcprmpw  12857  1arithlem4  12889  ctiunctlemfo  13010  elrestr  13280  lss1d  14347  lspsn  14380  znf1o  14615  restopnb  14855  mopnex  15179  metrest  15180  mpodvdsmulf1o  15664  lgsquadlem1  15756  2sqlem2  15794  mul2sq  15795  2sqlem3  15796  2sqlem9  15803  2sqlem10  15804  nnnninfex  16388