Theorem stdpc5 1054

Description: An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis (ph -> A.xph) can be thought of as emulating "x is not free in ph." With this convention, 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 hbequid 1165. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5.

Hypothesis

Ref	Expression
stdpc5.1	|- (ph -> A.xph)

Assertion

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

Proof of Theorem stdpc5

Step	Hyp	Ref	Expression
1		stdpc5.1	. . 3 |- (ph -> A.xph)
2	1	19.21 1052	. 2 |- (A.x(ph -> ps) <-> (ph -> A.xps))
3	2	biimp 151	1 |- (A.x(ph -> ps) -> (ph -> A.xps))

Syntax hints:   -> wi 3  A.wal 951

This theorem is referenced by:  ra5 1990

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972  ax-6o 975

This theorem depends on definitions:  df-bi 147

Copyright terms: Public domain