Theorem r19.23ai 1283

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 1206	. 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 277	. . 3 |- ((x e. A /\ ph) -> ps)
5	2, 4	19.23ai 746	. 2 |- (E.x(x e. A /\ ph) -> ps)
6	1, 5	sylbi 174	1 |- (E.x e. A ph -> ps)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  r19.23aiv 1284  tfinds 2401  r1val1 3502  rankuni 3533

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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