Theorem rspcedeq1vd 2839

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2836 for equations, in which the left 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
rspcedeq1vd	|- ( ph -> E. x e. B C = D )

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

Allowed substitution hint:   C( x)

Proof of Theorem rspcedeq1vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	. 2 |- ( ph -> A e. B )
2		rspcedeqvd.2	. . 3 |- ( ( ph /\ x = A ) -> C = D )
3	2	eqeq1d 2174	. 2 |- ( ( ph /\ x = A ) -> ( C = D <-> D = D ) )
4		eqidd 2166	. 2 |- ( ph -> D = D )
5	1, 3, 4	rspcedvd 2836	1 |- ( ph -> E. x e. B C = D )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   E.wrex 2445

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728

This theorem is referenced by: (None)