Theorem nfa1w 42635

Description: Replace ax-10 2142 in nfa1 2152 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 2036	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
3		nfv 1914	. 2 ⊢ Ⅎ𝑥∀𝑦𝜓
4	2, 3	nfxfr 1853	1 ⊢ Ⅎ𝑥∀𝑥𝜑

Syntax hints:   ↔ wb 206  ∀wal 1538  Ⅎwnf 1783

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  df-nf 1784

This theorem is referenced by:  eu6w  42636