Theorem rspcedeq2vd 2921

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2917 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 2237	. . 3 |- ( ( ph /\ x = A ) -> D = C )
4	3	eqeq2d 2243	. 2 |- ( ( ph /\ x = A ) -> ( C = D <-> C = C ) )
5		eqidd 2232	. 2 |- ( ph -> C = C )
6	1, 4, 5	rspcedvd 2917	1 |- ( ph -> E. x e. B C = D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   E.wrex 2512

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805

This theorem is referenced by:  elpr2elpr  3864