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Theorem spim 1975
 Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1975 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 xψ
spim.2 (x = y → (φψ))
Assertion
Ref Expression
spim (xφψ)

Proof of Theorem spim
StepHypRef Expression
1 spim.1 . 2 xψ
2 spim.2 . . 3 (x = y → (φψ))
32ax-gen 1546 . 2 x(x = y → (φψ))
4 spimt 1974 . 2 ((Ⅎxψ x(x = y → (φψ))) → (xφψ))
51, 3, 4mp2an 653 1 (xφψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  spime  1976  chvar  1986  spimv  1990
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