Theorem stdpc5 1584

Description: An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis F/ x ph can be thought of as emulating " x is not free in ph". With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example x would not (for us) be free in x = x by nfequid 1702. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypothesis

Ref	Expression
stdpc5.1	|- F/ x ph

Assertion

Ref	Expression
stdpc5	|- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) )

Proof of Theorem stdpc5

Step	Hyp	Ref	Expression
1		stdpc5.1	. . 3 |- F/ x ph
2	1	19.21 1583	. 2 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
3	2	biimpi 120	1 |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) )

Syntax hints:   -> wi 4   A.wal 1351   F/wnf 1460

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 1447  ax-gen 1449  ax-4 1510  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  sbalyz  1999  ra5  3051