Theorem r19.21bi 1266

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

Hypothesis

Ref	Expression
r19.21bi.1	|- (ph -> A.x e. A ps)

Assertion

Ref	Expression
r19.21bi	|- ((ph /\ x e. A) -> ps)

Proof of Theorem r19.21bi

Step	Hyp	Ref	Expression
1		r19.21bi.1	. . . 4 |- (ph -> A.x e. A ps)
2		df-ral 1205	. . . 4 |- (A.x e. A ps <-> A.x(x e. A -> ps))
3	1, 2	sylib 173	. . 3 |- (ph -> A.x(x e. A -> ps))
4	3	19.21bi 742	. 2 |- (ph -> (x e. A -> ps))
5	4	imp 277	1 |- ((ph /\ x e. A) -> ps)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  rspec2 1267  rspec3 1268  r19.21be 1269  prcdpq 3891  prnmax 3893

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

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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