Theorem ra5 3131

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 1798. (Contributed by NM, 16-Jan-2004.)

Hypothesis

Ref	Expression
ra5.1	|- F/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 2620	. . . 4 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
2		bi2.04 350	. . . . 5 |- ((x e. A -> (ph -> ps)) <-> (ph -> (x e. A -> ps)))
3	2	albii 1566	. . . 4 |- (A.x(x e. A -> (ph -> ps)) <-> A.x(ph -> (x e. A -> ps)))
4	1, 3	bitri 240	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(ph -> (x e. A -> ps)))
5		ra5.1	. . . 4 |- F/xph
6	5	stdpc5 1798	. . 3 |- (A.x(ph -> (x e. A -> ps)) -> (ph -> A.x(x e. A -> ps)))
7	4, 6	sylbi 187	. 2 |- (A.x e. A (ph -> ps) -> (ph -> A.x(x e. A -> ps)))
8		df-ral 2620	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
9	7, 8	syl6ibr 218	1 |- (A.x e. A (ph -> ps) -> (ph -> A.x e. A ps))

Syntax hints:   -> wi 4  A.wal 1540   F/wnf 1544   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by: (None)