Theorem rax5 1472

Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 59. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 739.

Hypothesis

Ref	Expression
rax5.1	|- (ph -> A.xph)

Assertion

Ref	Expression
rax5	|- (A.x e. A (ph -> ps) -> (ph -> A.x e. A ps))

Proof of Theorem rax5

Step	Hyp	Ref	Expression
1		df-ral 1205	. . . 4 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
2		bi2.04 141	. . . . 5 |- ((x e. A -> (ph -> ps)) <-> (ph -> (x e. A -> ps)))
3	2	bial 695	. . . 4 |- (A.x(x e. A -> (ph -> ps)) <-> A.x(ph -> (x e. A -> ps)))
4	1, 3	bitr 151	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(ph -> (x e. A -> ps)))
5		rax5.1	. . . 4 |- (ph -> A.xph)
6	5	stdpc5 739	. . 3 |- (A.x(ph -> (x e. A -> ps)) -> (ph -> A.x(x e. A -> ps)))
7	4, 6	sylbi 174	. 2 |- (A.x e. A (ph -> ps) -> (ph -> A.x(x e. A -> ps)))
8		df-ral 1205	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
9	7, 8	syl6ibr 186	1 |- (A.x e. A (ph -> ps) -> (ph -> A.x e. A ps))

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

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-ral 1205

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