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Metamath Proof Explorer

Theorem naecoms

Description: A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jan-2002) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	naecoms.1	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )
	Assertion	naecoms	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		naecoms.1	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )
2		aecom	⊢ ( ∀ 𝑥 𝑥 = 𝑦 ↔ ∀ 𝑦 𝑦 = 𝑥 )
3	2 1	sylnbir	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → 𝜑 )