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Theorem spw 1965
Description: Weak version of the specialization scheme sp 2051. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2051 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2051 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2010 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2051 are spfw 1963 (minimal distinct variable requirements), spnfw 1926 (when 𝑥 is not free in ¬ 𝜑), spvw 1896 (when 𝑥 does not appear in 𝜑), sptruw 1731 (when 𝜑 is true), and spfalw 1927 (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 1837 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
2 ax-5 1837 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
3 ax-5 1837 . 2 𝜑 → ∀𝑦 ¬ 𝜑)
4 spw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 3, 4spfw 1963 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by:  hba1w  1972  hba1wOLD  1973  spaev  1976  ax12w  2008  bj-ssblem1  32605  bj-ax12w  32640
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