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Theorem spfalw 1670
Description: Version of sp 1716 when  ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)
Hypothesis
Ref Expression
spfalw.1  |-  -.  ph
Assertion
Ref Expression
spfalw  |-  ( A. x ph  ->  ph )

Proof of Theorem spfalw
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 spfalw.1 . . . 4  |-  -.  ph
21bifal 1318 . . 3  |-  ( ph  <->  F.  )
32a1i 10 . 2  |-  ( x  =  y  ->  ( ph 
<->  F.  ) )
43spw 1660 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    F. wfal 1308   A.wal 1527
This theorem is referenced by:  19.2  1671  ax9dgen  1690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-fal 1311  df-ex 1529
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