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Theorem hbth 1562
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
StepHypRef Expression
1 hbth.1 . . 3  |-  ph
21ax-gen 1556 . 2  |-  A. x ph
32a1i 11 1  |-  ( ph  ->  A. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550
This theorem is referenced by:  nfth  1563  spfalw  1685  spimehOLD  1841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-gen 1556
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