Theorem rspcedv 2872

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 158	. 2 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
4	1, 3	rspcimedv 2870	1 |- ( ph -> ( ch -> E. x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765

This theorem is referenced by:  rspcedvd  2874  rexxfrd  4498  enomnilem  7204  enmkvlem  7227  ltexnqq  7475  halfnqq  7477  ltbtwnnqq  7482  genpml  7584  genpmu  7585  genprndl  7588  genprndu  7589  axarch  7958  apreap  8614