Theorem nfa1w 42235

Description: Replace ax-10 2129 in nfa1 2140 with a substitution hypothesis. (Contributed by SN, 7-Jun-2025.)

Hypothesis

Ref	Expression
nfa1w.x	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
nfa1w	⊢ Ⅎ𝑥∀𝑥𝜑

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

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

Proof of Theorem nfa1w

Step	Hyp	Ref	Expression
1		alex 1820	. 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
2		exim 1828	. . . . . 6 ⊢ (∀𝑥(∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑) → (∃𝑥∃𝑥 ¬ 𝜑 → ∃𝑥∀𝑥∃𝑥 ¬ 𝜑))
3		df-ex 1774	. . . . . . 7 ⊢ (∃𝑥∀𝑥∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥∃𝑥 ¬ 𝜑)
4		nfa1w.x	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	notbid 317	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
6	5	cbvexvw 2032	. . . . . . . . . 10 ⊢ (∃𝑥 ¬ 𝜑 ↔ ∃𝑦 ¬ 𝜓)
7	6	a1i 11	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑥 ¬ 𝜑 ↔ ∃𝑦 ¬ 𝜓))
8	7	hbn1w 2041	. . . . . . . 8 ⊢ (¬ ∀𝑥∃𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥∃𝑥 ¬ 𝜑)
9	8	con1i 147	. . . . . . 7 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
10	3, 9	sylbi 216	. . . . . 6 ⊢ (∃𝑥∀𝑥∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
11	2, 10	syl6 35	. . . . 5 ⊢ (∀𝑥(∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑) → (∃𝑥∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑))
12	11	nfd 1784	. . . 4 ⊢ (∀𝑥(∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑) → Ⅎ𝑥∃𝑥 ¬ 𝜑)
13	5	hbe1w 2043	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∀𝑥∃𝑥 ¬ 𝜑)
14	12, 13	mpg 1791	. . 3 ⊢ Ⅎ𝑥∃𝑥 ¬ 𝜑
15	14	nfn 1852	. 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥 ¬ 𝜑
16	1, 15	nfxfr 1847	1 ⊢ Ⅎ𝑥∀𝑥𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778

This theorem is referenced by:  eu6w  42236