Theorem hba1w 2048

Description: Weak version of hba1 2294. See comments for ax10w 2130. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)

Hypothesis

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

Assertion

Ref	Expression
hba1w	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

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

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

Proof of Theorem hba1w

Step	Hyp	Ref	Expression
1		hbn1w.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cbvalvw 2036	. . . . . 6 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
3	2	notbii 320	. . . . 5 ⊢ (¬ ∀𝑥𝜑 ↔ ¬ ∀𝑦𝜓)
4	3	a1i 11	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ ∀𝑥𝜑 ↔ ¬ ∀𝑦𝜓))
5	4	spw 2034	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
6	5	con2i 139	. 2 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
7	4	hbn1w 2047	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
8	1	hbn1w 2047	. . . 4 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
9	8	con1i 147	. . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
10	9	alimi 1811	. 2 ⊢ (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
11	6, 7, 10	3syl 18	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  exexw  2052  ralidmw  4488