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Theorem spim 1957
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1957 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.) (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 a9e 1952 . . 3  |-  E. x  x  =  y
3 spim.2 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
42, 3eximii 1587 . 2  |-  E. x
( ph  ->  ps )
51, 419.36i 1893 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553
This theorem is referenced by:  spimeOLD  1959  spimv  1963  chvar  1968  cbv3  1971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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