Theorem hbth 1456

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. x ph ) 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. x ph )

Proof of Theorem hbth

Step	Hyp	Ref	Expression
1		hbth.1	. . 3 |- ph
2	1	ax-gen 1442	. 2 |- A. x ph
3	2	a1i 9	1 |- ( ph -> A. x ph )

Syntax hints:   -> wi 4   A.wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-gen 1442

This theorem is referenced by:  nfth  1457  sbieh  1783  bj-sbimeh  13807