This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) Avoid ax-12 . (Revised by GG, 12-Oct-2024) (Proof shortened by Wolf Lammen, 22-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ceqsexv.1	$⊢ A \in V$
2		ceqsexv.2	$⊢ x = A \to (φ \leftrightarrow ψ)$
3		alinexa	$⊢ \forall x (x = A \to \neg φ) \leftrightarrow \neg \exists x (x = A \land φ)$
4	2	notbid	$⊢ x = A \to (\neg φ \leftrightarrow \neg ψ)$
5	1 4	ceqsalv	$⊢ \forall x (x = A \to \neg φ) \leftrightarrow \neg ψ$
6	3 5	bitr3i	$⊢ \neg \exists x (x = A \land φ) \leftrightarrow \neg ψ$
7	6	con4bii	$⊢ \exists x (x = A \land φ) \leftrightarrow ψ$