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Metamath Proof Explorer

Theorem bj-aecomsv

Description: Version of aecoms with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms should not be very useful since -. A. x x = y , DV ( x , y ) is true when the universe has at least two objects (see dtru ). (Contributed by BJ, 31-May-2019) (Proof modification is discouraged.)

		Ref	Expression
	Hypothesis	bj-aecomsv.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )
	Assertion	bj-aecomsv	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-aecomsv.1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )
2		aevlem	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ∀ 𝑥 𝑥 = 𝑦 )
3	2 1	syl	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → 𝜑 )