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Metamath Proof Explorer

Theorem ceqsexv2d

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025) (Proof shortened by SN, 5-Jun-2025)

	Ref	Expression
Hypotheses	ceqsexv2d.1	$⊢ A \in V$
	ceqsexv2d.2	$⊢ x = A \to (φ \leftrightarrow ψ)$
	ceqsexv2d.3	$⊢ ψ$
Assertion	ceqsexv2d	$⊢ \exists x φ$

Proof

Step	Hyp	Ref	Expression
1		ceqsexv2d.1	$⊢ A \in V$
2		ceqsexv2d.2	$⊢ x = A \to (φ \leftrightarrow ψ)$
3		ceqsexv2d.3	$⊢ ψ$
4	1	isseti	$⊢ \exists x x = A$
5	3 2	mpbiri	$⊢ x = A \to φ$
6	4 5	eximii	$⊢ \exists x φ$