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Theorem a4im 1867
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The a4im 1867 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
a4im.1  |-  F/ x ps
a4im.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
a4im  |-  ( A. x ph  ->  ps )

Proof of Theorem a4im
StepHypRef Expression
1 a4im.1 . 2  |-  F/ x ps
2 a4im.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
32ax-gen 1536 . 2  |-  A. x
( x  =  y  ->  ( ph  ->  ps ) )
4 a4imt 1866 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )  ->  ( A. x ph  ->  ps ) )
51, 3, 4mp2an 656 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532   F/wnf 1539    = wceq 1619
This theorem is referenced by:  a4ime  1868  chvar  1878  a4imv  1922
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-nf 1540
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