Theorem ra5 3122

Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1633. (Contributed by NM, 16-Jan-2004.)

Hypothesis

Ref	Expression
ra5.1	|- F/ x ph

Assertion

Ref	Expression
ra5	|- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) )

Proof of Theorem ra5

Step	Hyp	Ref	Expression
1		df-ral 2516	. . . 4 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		bi2.04 248	. . . . 5 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ph -> ( x e. A -> ps ) ) )
3	2	albii 1519	. . . 4 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
4	1, 3	bitri 184	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
5		ra5.1	. . . 4 |- F/ x ph
6	5	stdpc5 1633	. . 3 |- ( A. x ( ph -> ( x e. A -> ps ) ) -> ( ph -> A. x ( x e. A -> ps ) ) )
7	4, 6	sylbi 121	. 2 |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x ( x e. A -> ps ) ) )
8		df-ral 2516	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
9	7, 8	imbitrrdi 162	1 |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   A.wal 1396   F/wnf 1509   e. wcel 2202   A.wral 2511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516

This theorem is referenced by: (None)