Theorem ra5 2539

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 1405.

Hypothesis

Ref	Expression
ra5.1	|- (ph -> A.xph)

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 2109	. . . 4 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
2		bi2.04 177	. . . . 5 |- ((x e. A -> (ph -> ps)) <-> (ph -> (x e. A -> ps)))
3	2	albii 1346	. . . 4 |- (A.x(x e. A -> (ph -> ps)) <-> A.x(ph -> (x e. A -> ps)))
4	1, 3	bitri 190	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(ph -> (x e. A -> ps)))
5		ra5.1	. . . 4 |- (ph -> A.xph)
6	5	stdpc5 1405	. . 3 |- (A.x(ph -> (x e. A -> ps)) -> (ph -> A.x(x e. A -> ps)))
7	4, 6	sylbi 216	. 2 |- (A.x e. A (ph -> ps) -> (ph -> A.x(x e. A -> ps)))
8		df-ral 2109	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
9	7, 8	syl6ibr 230	1 |- (A.x e. A (ph -> ps) -> (ph -> A.x e. A ps))

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  A.wral 2105

This theorem is referenced by:  r19.21 13818  wfr3g 13956  frr3g 13980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-an 242  df-ral 2109

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