Theorem rspceeqv 2895

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 2217	. 2 |- ( x = A -> ( E = C <-> E = D ) )
3	2	rspcev 2877	1 |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   E.wrex 2485

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774

This theorem is referenced by:  elixpsn  6822  ixpsnf1o  6823  elfir  7075  0ct  7209  ctmlemr  7210  ctssdclemn0  7212  fodju0  7249  mertenslemi1  11846  mertenslem2  11847  nninfctlemfo  12361  pcprmpw  12657  1arithlem4  12689  ctiunctlemfo  12810  elrestr  13079  lss1d  14145  lspsn  14178  znf1o  14413  restopnb  14653  mopnex  14977  metrest  14978  mpodvdsmulf1o  15462  lgsquadlem1  15554  2sqlem2  15592  mul2sq  15593  2sqlem3  15594  2sqlem9  15601  2sqlem10  15602  nnnninfex  15959