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Metamath Proof Explorer

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 GG, 18-Nov-2024)

	Ref	Expression
Hypotheses	equsalv.nf	$⊢ Ⅎ x ψ$
	equsalv.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	equsexv	$⊢ \exists x (x = y \land φ) \leftrightarrow ψ$

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	$⊢ Ⅎ x ψ$
2		equsalv.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
3	2	biimpa	$⊢ (x = y \land φ) \to ψ$
4	1 3	exlimi	$⊢ \exists x (x = y \land φ) \to ψ$
5	1 2	equsalv	$⊢ \forall x (x = y \to φ) \leftrightarrow ψ$
6		equs4v	$⊢ \forall x (x = y \to φ) \to \exists x (x = y \land φ)$
7	5 6	sylbir	$⊢ ψ \to \exists x (x = y \land φ)$
8	4 7	impbii	$⊢ \exists x (x = y \land φ) \leftrightarrow ψ$