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Theorem cbvaliw 2009
 Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
cbvaliw.1 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
cbvaliw.2 𝜓 → ∀𝑥 ¬ 𝜓)
cbvaliw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvaliw (∀𝑥𝜑 → ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvaliw
StepHypRef Expression
1 cbvaliw.1 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 cbvaliw.2 . . 3 𝜓 → ∀𝑥 ¬ 𝜓)
3 cbvaliw.3 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimw 1969 . 2 (∀𝑥𝜑𝜓)
51, 4alrimih 1820 1 (∀𝑥𝜑 → ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-6 1966 This theorem depends on definitions:  df-bi 209  df-ex 1777 This theorem is referenced by:  spfw  2036  cbvalw  2038
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