Theorem rspesbca 3127

Description: Existence form of rspsbca 3126. (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 3050	. . 3 |- (y = A -> ([y / x]ph <-> [.A / x ].ph))
2	1	rspcev 2956	. 2 |- ((A e. B /\ [.A / x ].ph) -> E.y e. B [y / x]ph)
3		cbvrexsv 2848	. 2 |- (E.x e. B ph <-> E.y e. B [y / x]ph)
4	2, 3	sylibr 203	1 |- ((A e. B /\ [.A / x ].ph) -> E.x e. B ph)

Syntax hints:   -> wi 4   /\ wa 358  [wsb 1648   e. wcel 1710  E.wrex 2616   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048

This theorem is referenced by:  spesbc  3128