Theorem spw 1954

Description: Weak version of the specialization scheme sp 2041. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2041 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2041 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1999 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2041 are spfw 1952 (minimal distinct variable requirements), spnfw 1915 (when 𝑥 is not free in ¬ 𝜑), spvw 1885 (when 𝑥 does not appear in 𝜑), sptruw 1724 (when 𝜑 is true), and spfalw 1916 (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 1827	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1827	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1827	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 1952	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  hba1w  1961  hba1wOLD  1962  spaev  1965  ax12w  1997  bj-ssblem1  31613  bj-ax12w  31646