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Theorem spfw 1704
Description: Weak version of sp 1764. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
spfw.1  |-  ( -. 
ps  ->  A. x  -.  ps )
spfw.2  |-  ( A. x ph  ->  A. y A. x ph )
spfw.3  |-  ( -. 
ph  ->  A. y  -.  ph )
spfw.4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spfw  |-  ( A. x ph  ->  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem spfw
StepHypRef Expression
1 spfw.2 . . 3  |-  ( A. x ph  ->  A. y A. x ph )
2 ax-5 1567 . . 3  |-  ( A. y ( A. x ph  ->  ps )  -> 
( A. y A. x ph  ->  A. y ps ) )
3 spfw.3 . . . 4  |-  ( -. 
ph  ->  A. y  -.  ph )
4 spfw.4 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
54biimprd 216 . . . . 5  |-  ( x  =  y  ->  ( ps  ->  ph ) )
65equcoms 1694 . . . 4  |-  ( y  =  x  ->  ( ps  ->  ph ) )
73, 6spimw 1681 . . 3  |-  ( A. y ps  ->  ph )
81, 2, 7syl56 33 . 2  |-  ( A. y ( A. x ph  ->  ps )  -> 
( A. x ph  ->  ph ) )
9 spfw.1 . . 3  |-  ( -. 
ps  ->  A. x  -.  ps )
104biimpd 200 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
119, 10spimw 1681 . 2  |-  ( A. x ph  ->  ps )
128, 11mpg 1558 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   A.wal 1550
This theorem is referenced by:  spnfwOLD  1705  spw  1707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
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