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Theorem spfw 2007
Description: Weak version of sp 2091. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
Hypotheses
Ref Expression
spfw.1 𝜓 → ∀𝑥 ¬ 𝜓)
spfw.2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
spfw.3 𝜑 → ∀𝑦 ¬ 𝜑)
spfw.4 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spfw (∀𝑥𝜑𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spfw
StepHypRef Expression
1 spfw.2 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 spfw.1 . . 3 𝜓 → ∀𝑥 ¬ 𝜓)
3 spfw.4 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43biimpd 219 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbvaliw 1979 . 2 (∀𝑥𝜑 → ∀𝑦𝜓)
6 spfw.3 . . 3 𝜑 → ∀𝑦 ¬ 𝜑)
73biimprd 238 . . . 4 (𝑥 = 𝑦 → (𝜓𝜑))
87equcoms 1993 . . 3 (𝑦 = 𝑥 → (𝜓𝜑))
96, 8spimw 1972 . 2 (∀𝑦𝜓𝜑)
105, 9syl 17 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  spw  2009
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