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Theorem spfw 1657
Description: Weak version of sp 1716. 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 1544 . . 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 214 . . . . 5  |-  ( x  =  y  ->  ( ps  ->  ph ) )
65equcoms 1651 . . . 4  |-  ( y  =  x  ->  ( ps  ->  ph ) )
73, 6spimw 1638 . . 3  |-  ( A. y ps  ->  ph )
81, 2, 7syl56 30 . 2  |-  ( A. y ( A. x ph  ->  ps )  -> 
( A. x ph  ->  ph ) )
9 spfw.1 . . 3  |-  ( -. 
ps  ->  A. x  -.  ps )
104biimpd 198 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
119, 10spimw 1638 . 2  |-  ( A. x ph  ->  ps )
128, 11mpg 1535 1  |-  ( A. x ph  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  spnfwOLD  1658
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-ex 1529
  Copyright terms: Public domain W3C validator