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 → (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 1837	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1837	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1837	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 1963	1 ⊢ (∀𝑥𝜑 → 𝜑)

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