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Theorem spfalw 1679
Description: Version of sp 1755 when  ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
Hypothesis
Ref Expression
spfalw.1  |-  -.  ph
Assertion
Ref Expression
spfalw  |-  ( A. x ph  ->  ph )

Proof of Theorem spfalw
StepHypRef Expression
1 spfalw.1 . . 3  |-  -.  ph
21hbth 1558 . 2  |-  ( -. 
ph  ->  A. x  -.  ph )
32spnfw 1677 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546
This theorem is referenced by:  19.2OLD  1707  ax9dgen  1723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-9 1661
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
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