Theorem stdpc5 739

Description: An axiom of standard predicate calculus. Axiom 5 of [Mendelson] p. 59. The hypothesis (ph -> A.xph) can be thought of as "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 since (by eqid 810 and hbth 697) we can prove (x = x -> A.xx = x).

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	19.21 738	. 2 |- (A.x(ph -> ps) <-> (ph -> A.xps))
3	2	biimp 133	1 |- (A.x(ph -> ps) -> (ph -> A.xps))

Syntax hints:   -> wi 2  A.wal 672

This theorem is referenced by:  rax5 1472

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128

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