Theorem equsexhv

Description: An equivalence related to implicit substitution. Version of equsexh with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	equsalhw.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
	equsalhw.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsexhv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsalhw.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
2		equsalhw.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	1	nf5i	⊢ Ⅎ 𝑥 𝜓
4	3 2	equsexv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )