Theorem hbth 1003

Description: No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form |- (ph -> A.xph) from smaller formulas of this form. These are useful for constructing hypotheses that state "x is (effectively) not free in ph."

Hypothesis

Ref	Expression
hbth.1	|- ph

Assertion

Ref	Expression
hbth	|- (ph -> A.xph)

Proof of Theorem hbth

Step	Hyp	Ref	Expression
1		hbth.1	. . 3 |- ph
2	1	ax-gen 965	. 2 |- A.xph
3	2	a1i 8	1 |- (ph -> A.xph)

Syntax hints:   -> wi 3  A.wal 956

This theorem is referenced by:  sbie 1198  a12lem1 1378  ralbii 1670  rexbii 1671  sbcralg 1997  sbcrexg 1998  infcvgaux1 7219

This theorem was proved from axioms:  ax-1 4  ax-mp 7  ax-gen 965

Copyright terms: Public domain