Theorem ssrexv 2118

Description: Existential quantification restricted to a subclass.

Assertion

Ref	Expression
ssrexv	|- (A (_ B -> (E.x e. A ph -> E.x e. B ph))

Distinct variable groups:   x,A   x,B

Proof of Theorem ssrexv

Step	Hyp	Ref	Expression
1		ssel 2066	. . 3 |- (A (_ B -> (x e. A -> x e. B))
2	1	anim1d 562	. 2 |- (A (_ B -> ((x e. A /\ ph) -> (x e. B /\ ph)))
3	2	r19.22dv2 1739	1 |- (A (_ B -> (E.x e. A ph -> E.x e. B ph))

Syntax hints:   -> wi 3   e. wcel 960  E.wrex 1649   (_ wss 2050

This theorem is referenced by:  clmi1 7086  ivthlem7 7287  bastop 7641  efifolem4 8720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-in 2054  df-ss 2056

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