Theorem nfa1w 42778

Description: Replace ax-10 2144 in nfa1 2154 with a substitution hypothesis. (Contributed by SN, 2-Sep-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		nfa1w.x	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cbvalvw 2037	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
3		nfv 1915	. 2 ⊢ Ⅎ𝑥∀𝑦𝜓
4	2, 3	nfxfr 1854	1 ⊢ Ⅎ𝑥∀𝑥𝜑

Syntax hints:   ↔ wb 206  ∀wal 1539  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by:  eu6w  42779