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Theorem spw 2009
Description: Weak version of the specialization scheme sp 2091. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2091 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2091 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2052 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2091 are spfw 2007 (minimal distinct variable requirements), spnfw 1974 (when 𝑥 is not free in ¬ 𝜑), spvw 1955 (when 𝑥 does not appear in 𝜑), sptruw 1773 (when 𝜑 is true), and spfalw 1975 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
Hypothesis
Ref Expression
spw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spw (∀𝑥𝜑𝜑)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw
StepHypRef Expression
1 ax-5 1879 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
2 ax-5 1879 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
3 ax-5 1879 . 2 𝜑 → ∀𝑦 ¬ 𝜑)
4 spw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 3, 4spfw 2007 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:  hba1w  2016  hba1wOLD  2017  spaev  2020  ax12w  2050  bj-ssblem1  32755  bj-ax12w  32790
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