Theorem rspcedeq2vd 2826

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2822 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)

Hypotheses

Ref	Expression
rspcedeqvd.1	|- ( ph -> A e. B )
rspcedeqvd.2	|- ( ( ph /\ x = A ) -> C = D )

Assertion

Ref	Expression
rspcedeq2vd	|- ( ph -> E. x e. B C = D )

Distinct variable groups:   x, A   x, B   ph, x   x, C

Allowed substitution hint:   D( x)

Proof of Theorem rspcedeq2vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	. 2 |- ( ph -> A e. B )
2		rspcedeqvd.2	. . . 4 |- ( ( ph /\ x = A ) -> C = D )
3	2	eqcomd 2163	. . 3 |- ( ( ph /\ x = A ) -> D = C )
4	3	eqeq2d 2169	. 2 |- ( ( ph /\ x = A ) -> ( C = D <-> C = C ) )
5		eqidd 2158	. 2 |- ( ph -> C = C )
6	1, 4, 5	rspcedvd 2822	1 |- ( ph -> E. x e. B C = D )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   E.wrex 2436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714

This theorem is referenced by: (None)