Theorem rspcedv 2829

Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
rspcedv	|- ( 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 rspcedv

Step	Hyp	Ref	Expression
1		rspcdv.1	. 2 |- ( ph -> A e. B )
2		rspcdv.2	. . 3 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	2	biimprd 157	. 2 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
4	1, 3	rspcimedv 2827	1 |- ( ph -> ( ch -> E. x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   E.wrex 2443

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723

This theorem is referenced by:  rspcedvd  2831  rexxfrd  4435  enomnilem  7093  enmkvlem  7116  ltexnqq  7340  halfnqq  7342  ltbtwnnqq  7347  genpml  7449  genpmu  7450  genprndl  7453  genprndu  7454  axarch  7823  apreap  8476