Theorem hbth 1552

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." (Contributed by NM, 5-Aug-1993.)

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 1546	. 2 |- A.xph
3	2	a1i 10	1 |- (ph -> A.xph)

Syntax hints:   -> wi 4  A.wal 1540

This theorem was proved from axioms:  ax-1 5  ax-mp 8  ax-gen 1546

This theorem is referenced by:  nfth  1553  spfalw  1672  spimehOLD  1821