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Theorem spimvw 2000
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 2389 and spimfv 2233. (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 1914 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
2 spimvw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2spimw 1975 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972
This theorem depends on definitions:  df-bi 206  df-ex 1783
This theorem is referenced by:  spvv  2001  cbvalivw  2011  alcomiw  2047  axc16i  2435  ax9ALT  2732  reu6  3689  disj  4412  elALT2  5329  fvn0ssdmfun  7030  aev-o  37422  axc11next  42760  funressnvmo  45353
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