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Theorem spim 1928
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1928 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
spim.1  |-  F/ x ps
spim.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spim  |-  ( A. x ph  ->  ps )

Proof of Theorem spim
StepHypRef Expression
1 spim.1 . 2  |-  F/ x ps
2 spim.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
32ax-gen 1536 . 2  |-  A. x
( x  =  y  ->  ( ph  ->  ps ) )
4 spimt 1927 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )  ->  ( A. x ph  ->  ps ) )
51, 3, 4mp2an 653 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   F/wnf 1534
This theorem is referenced by:  spime  1929  chvar  1939  spimv  1943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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