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  6824  ixpsnf1o  6825  elfir  7077  0ct  7211  ctmlemr  7212  ctssdclemn0  7214  fodju0  7251  mertenslemi1  11879  mertenslem2  11880  nninfctlemfo  12394  pcprmpw  12690  1arithlem4  12722  ctiunctlemfo  12843  elrestr  13112  lss1d  14178  lspsn  14211  znf1o  14446  restopnb  14686  mopnex  15010  metrest  15011  mpodvdsmulf1o  15495  lgsquadlem1  15587  2sqlem2  15625  mul2sq  15626  2sqlem3  15627  2sqlem9  15634  2sqlem10  15635  nnnninfex  15996