Theorem stdpc5 2076

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 1940. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1867 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)

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 2075	. 2 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
3	2	biimpi 206	1 |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) )

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  2stdpc5  32816