Theorem r19.23ai 1739

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.)

Hypotheses

Ref	Expression
r19.23ai.1	|- (ps -> A.xps)
r19.23ai.2	|- (x e. A -> (ph -> ps))

Assertion

Ref	Expression
r19.23ai	|- (E.x e. A ph -> ps)

Proof of Theorem r19.23ai

Step	Hyp	Ref	Expression
1		df-rex 1647	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		r19.23ai.1	. . 3 |- (ps -> A.xps)
3		r19.23ai.2	. . . 4 |- (x e. A -> (ph -> ps))
4	3	imp 350	. . 3 |- ((x e. A /\ ph) -> ps)
5	2, 4	19.23ai 1062	. 2 |- (E.x(x e. A /\ ph) -> ps)
6	1, 5	sylbi 199	1 |- (E.x e. A ph -> ps)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   e. wcel 956  E.wex 978  E.wrex 1643

This theorem is referenced by:  r19.23aiv 1740  tfinds 3156  r1val1 4638  rankuni2 4670

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973  ax-6o 976

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-rex 1647

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