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Metamath Proof Explorer

Theorem equs4v

Description: Version of equs4 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-May-1993) (Revised by BJ, 31-May-2019)

		Ref	Expression
	Assertion	equs4v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
2		exintr	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) )
3	1 2	mpi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )