Theorem hba1wOLD 1962

Description: Obsolete proof of hba1w 1961 as of 10-Oct-2021. (Contributed by NM, 9-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
hbn1w.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
hba1wOLD	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

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

Proof of Theorem hba1wOLD

Step	Hyp	Ref	Expression
1		hbn1w.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cbvalvw 1956	. . . . . 6 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
3	2	a1i 11	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
4	3	notbid 307	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ ∀𝑥𝜑 ↔ ¬ ∀𝑦𝜓))
5	4	spw 1954	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
6	5	con2i 133	. 2 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
7	4	hbn1w 1960	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
8	1	hbn1w 1960	. . . 4 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
9	8	con1i 143	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
10	9	alimi 1730	. 2 ⊢ (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
11	6, 7, 10	3syl 18	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

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: (None)