Theorem rspesbca 3070

Description: Existence form of rspsbca 3069. (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 2988	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	rspcev 2864	. 2 |- ( ( A e. B /\ [. A / x ]. ph ) -> E. y e. B [ y / x ] ph )
3		cbvrexsv 2743	. 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 1773   e. wcel 2164   E.wrex 2473   [.wsbc 2985

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986

This theorem is referenced by:  spesbc  3071