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Theorem spimvw 2002
 Description: A weak form of specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. For stronger forms using more axioms, see spimv 2397 and spimfv 2239. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
spimvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimvw (∀𝑥𝜑𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem spimvw
StepHypRef Expression
1 ax-5 1911 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
2 spimvw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2spimw 1973 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-5 1911  ax-6 1970 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  spvv  2003  cbvalivw  2014  alcomiw  2050  alcomiwOLD  2051  axc16i  2447  ax9ALT  2794  reu6  3665  disj  4355  el  5235  fvn0ssdmfun  6819  aev-o  36243  axc11next  41125  funressnvmo  43652
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