Theorem ssrexv 2115

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 2063	. . 3 |- (A (_ B -> (x e. A -> x e. B))
2	1	anim1d 560	. 2 |- (A (_ B -> ((x e. A /\ ph) -> (x e. B /\ ph)))
3	2	r19.22dv2 1736	1 |- (A (_ B -> (E.x e. A ph -> E.x e. B ph))

Syntax hints:   -> wi 3   e. wcel 958  E.wrex 1646   (_ wss 2047

This theorem is referenced by:  clmi1 7086  ivthlem7 7287  bastop 7642  efifolem4 8725

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-in 2051  df-ss 2053

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