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Metamath Proof Explorer

Theorem nfich1

Description: The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023)

		Ref	Expression
	Assertion	nfich1	⊢ Ⅎ 𝑥 [ 𝑥 ⇄ 𝑦 ] 𝜑

Step	Hyp	Ref	Expression
1		df-ich	⊢ ( [ 𝑥 ⇄ 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 ) )
2		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
3	1 2	nfxfr	⊢ Ⅎ 𝑥 [ 𝑥 ⇄ 𝑦 ] 𝜑