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Theorem spimw 1973
 Description: Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
Hypotheses
Ref Expression
spimw.1 𝜓 → ∀𝑥 ¬ 𝜓)
spimw.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimw (∀𝑥𝜑𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimw
StepHypRef Expression
1 ax6v 1971 . 2 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 spimw.1 . . 3 𝜓 → ∀𝑥 ¬ 𝜓)
3 spimw.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimfw 1968 . 2 (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑𝜓))
51, 4ax-mp 5 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-6 1970 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  spnfw  1984  spimvw  2002  cbvaliw  2013  spfw  2040
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