Theorem rspesbca 3047

Description: Existence form of rspsbca 3046. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspesbca	|- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph )

Distinct variable group:   x, B

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem rspesbca

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2965	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	rspcev 2841	. 2 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. y e. B [ y / x ] ph )
3		cbvrexsv 2720	. 2 |- ( E. x e. B ph <-> E. y e. B [ y / x ] ph )
4	2, 3	sylibr 134	1 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph )

Syntax hints:   -> wi 4   /\ wa 104   [wsb 1762   e. wcel 2148   E.wrex 2456   [.wsbc 2962

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963

This theorem is referenced by:  spesbc  3048