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 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1879	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1879	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1879	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2007	1 ⊢ (∀𝑥𝜑 → 𝜑)

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