Theorem spfw 2066

Description: Weak version of sp 2222. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)

Hypotheses

Ref	Expression
spfw.1	⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
spfw.2	⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
spfw.3	⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
spfw.4	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spfw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem spfw

Step	Hyp	Ref	Expression
1		spfw.2	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		spfw.1	. . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
3		spfw.4	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	biimpd 232	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbvaliw 2039	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6		spfw.3	. . 3 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
7	3	biimprd 251	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
8	7	equcoms 2053	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
9	6, 8	spimw 2003	. 2 ⊢ (∀𝑦𝜓 → 𝜑)
10	5, 9	syl 18	1 ⊢ (∀𝑥𝜑 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  spw  2067