Theorem rspceeqv 2902

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 2219	. 2 |- ( x = A -> ( E = C <-> E = D ) )
3	2	rspcev 2884	1 |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   E.wrex 2487

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778

This theorem is referenced by:  elixpsn  6845  ixpsnf1o  6846  elfir  7101  0ct  7235  ctmlemr  7236  ctssdclemn0  7238  fodju0  7275  ccats1pfxeqrex  11206  mertenslemi1  11961  mertenslem2  11962  nninfctlemfo  12476  pcprmpw  12772  1arithlem4  12804  ctiunctlemfo  12925  elrestr  13194  lss1d  14260  lspsn  14293  znf1o  14528  restopnb  14768  mopnex  15092  metrest  15093  mpodvdsmulf1o  15577  lgsquadlem1  15669  2sqlem2  15707  mul2sq  15708  2sqlem3  15709  2sqlem9  15716  2sqlem10  15717  nnnninfex  16161