Theorem r19.21bi 1723

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 1647	. . . 4 |- (A.x e. A ps <-> A.x(x e. A -> ps))
3	1, 2	sylib 198	. . 3 |- (ph -> A.x(x e. A -> ps))
4	3	19.21bi 1059	. 2 |- (ph -> (x e. A -> ps))
5	4	imp 350	1 |- ((ph /\ x e. A) -> ps)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   e. wcel 957  A.wral 1643

This theorem is referenced by:  rspec2 1724  rspec3 1725  r19.21be 1726  prcdpq 5080  prnmax 5082  ubthlem3 8490

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1647

Copyright terms: Public domain