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Metamath Proof Explorer

Theorem equs4

Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition ( sbalex ) or a nonfreeness hypothesis ( equs45f ). Usage of this theorem is discouraged because it depends on ax-13 . See equs4v for a weaker version requiring fewer axioms. (Contributed by NM, 10-May-1993) (Proof shortened by Mario Carneiro, 20-May-2014) (Proof shortened by Wolf Lammen, 5-Feb-2018) (New usage is discouraged.)

		Ref	Expression
	Assertion	equs4	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax6e	⊢ ∃ 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 ) )