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Metamath Proof Explorer

Theorem nfcvf

Description: If x and y are distinct, then x is not free in y . Usage of this theorem is discouraged because it depends on ax-13 . See nfcv for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-ext . (Revised by Wolf Lammen, 10-May-2023) (New usage is discouraged.)

		Ref	Expression
	Assertion	nfcvf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑤 ¬ ∀ 𝑥 𝑥 = 𝑦
2		nfv	⊢ Ⅎ 𝑥 𝑤 ∈ 𝑧
3		elequ2	⊢ ( 𝑧 = 𝑦 → ( 𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑦 ) )
4	2 3	dvelimnf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑤 ∈ 𝑦 )
5	1 4	nfcd	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 )