Theorem spcimedv 2708

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimdv.1	|- ( ph -> A e. B )
spcimedv.2	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )

Assertion

Ref	Expression
spcimedv	|- ( ph -> ( ch -> E. x ps ) )

Distinct variable groups:   x, A   ph, x   ch, x

Allowed substitution hints:   ps( x)   B( x)

Proof of Theorem spcimedv

Step	Hyp	Ref	Expression
1		spcimedv.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
2	1	ex 114	. . 3 |- ( ph -> ( x = A -> ( ch -> ps ) ) )
3	2	alrimiv 1803	. 2 |- ( ph -> A. x ( x = A -> ( ch -> ps ) ) )
4		spcimdv.1	. 2 |- ( ph -> A e. B )
5		nfv 1467	. . 3 |- F/ x ch
6		nfcv 2229	. . 3 |- F/_ x A
7	5, 6	spcimegft 2700	. 2 |- ( A. x ( x = A -> ( ch -> ps ) ) -> ( A e. B -> ( ch -> E. x ps ) ) )
8	3, 4, 7	sylc 62	1 |- ( ph -> ( ch -> E. x ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1288   = wceq 1290   E.wex 1427   e. wcel 1439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624

This theorem is referenced by:  rspcimedv  2727  fihashf1rn  10260