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Theorem spim 2062
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2062 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
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 ax6e 2058 . . 3  |-  E. x  x  =  y
3 spim.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
42, 3eximii 1705 . 2  |-  E. x
( ph  ->  ps )
51, 419.36i 2022 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  spimv  2065  chvar  2069  cbv3  2071
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