Theorem spw 2037

Description: Weak version of the specialization scheme sp 2177. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2177 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2177 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2135 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2177 are spfw 2036 (minimal distinct variable requirements), spnfw 1980 (when 𝑥 is not free in ¬ 𝜑), spvw 1981 (when 𝑥 does not appear in 𝜑), sptruw 1803 (when 𝜑 is true), and spfalw 2000 (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 1907	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1907	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1907	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2036	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  hba1w  2050  spaev  2053  ax12w  2133  bj-ssblem1  33982  bj-ax12w  34005