Theorem equsexv

Description: An equivalence related to implicit substitution. Version of equsex with a disjoint variable condition, which does not require ax-13 . See equsexvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019) Avoid ax-10 . (Revised by Gino Giotto, 18-Nov-2024)

	Ref	Expression
Hypotheses	equsalv.nf	⊢ Ⅎ 𝑥 𝜓
	equsalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsexv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	⊢ Ⅎ 𝑥 𝜓
2		equsalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	2	biimpa	⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → 𝜓 )
4	1 3	exlimi	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → 𝜓 )
5	1 2	equsalv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )
6		equs4v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
7	5 6	sylbir	⊢ ( 𝜓 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
8	4 7	impbii	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )