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Metamath Proof Explorer

Theorem naecoms-o

Description: A commutation rule for distinct variable specifiers. Version of naecoms using ax-c11 . (Contributed by NM, 2-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nalequcoms-o.1	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )
2		aecom-o	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑦 = 𝑥 )
3	2 1	nsyl4	⊢ ( ¬ 𝜑 → ∀ 𝑦 𝑦 = 𝑥 )
4	3	con1i	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → 𝜑 )