Theorem rspcimedv 3311

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

Hypotheses

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

Assertion

Ref	Expression
rspcimedv	|- ( ph -> ( ch -> E. x e. B ps ) )

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

Allowed substitution hint:   ps( x)

Proof of Theorem rspcimedv

Step	Hyp	Ref	Expression
1		rspcimdv.1	. . . 4 |- ( ph -> A e. B )
2		rspcimedv.2	. . . . 5 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
3	2	con3d 148	. . . 4 |- ( ( ph /\ x = A ) -> ( -. ps -> -. ch ) )
4	1, 3	rspcimdv 3310	. . 3 |- ( ph -> ( A. x e. B -. ps -> -. ch ) )
5	4	con2d 129	. 2 |- ( ph -> ( ch -> -. A. x e. B -. ps ) )
6		dfrex2 2996	. 2 |- ( E. x e. B ps <-> -. A. x e. B -. ps )
7	5, 6	syl6ibr 242	1 |- ( ph -> ( ch -> E. x e. B ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  rspcedv  3313  scshwfzeqfzo  13572  symgfixfo  17859  slesolex  20488  usgr2pthlem  26659  clwlksfoclwwlk  26963