Theorem spfwOLD 1966

Description: Obsolete proof of spfw 1965 as of 10-Oct-2021. (Contributed by NM, 19-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

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
spfwOLD	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem spfwOLD

Step	Hyp	Ref	Expression
1		spfw.2	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		alim 1738	. . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) → (∀𝑦∀𝑥𝜑 → ∀𝑦𝜓))
3		spfw.3	. . . 4 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spfw.4	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	biimprd 238	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
6	5	equcoms 1947	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
7	3, 6	spimw 1926	. . 3 ⊢ (∀𝑦𝜓 → 𝜑)
8	1, 2, 7	syl56 36	. 2 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜑))
9		spfw.1	. . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
10	4	biimpd 219	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
11	9, 10	spimw 1926	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
12	8, 11	mpg 1724	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)