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Metamath Proof Explorer

Theorem exsb

Description: An equivalent expression for existence. One direction ( exsbim ) needs fewer axioms. (Contributed by NM, 2-Feb-2005) Avoid ax-13 . (Revised by Wolf Lammen, 16-Oct-2022)

		Ref	Expression
	Assertion	exsb	$⊢ \exists x φ \leftrightarrow \exists y \forall x (x = y \to φ)$

Proof

Step	Hyp	Ref	Expression
1		nfv	$⊢ Ⅎ y φ$
2		nfa1	$⊢ Ⅎ x \forall x (x = y \to φ)$
3		ax12v	$⊢ x = y \to (φ \to \forall x (x = y \to φ))$
4		sp	$⊢ \forall x (x = y \to φ) \to (x = y \to φ)$
5	4	com12	$⊢ x = y \to (\forall x (x = y \to φ) \to φ)$
6	3 5	impbid	$⊢ x = y \to (φ \leftrightarrow \forall x (x = y \to φ))$
7	1 2 6	cbvexv1	$⊢ \exists x φ \leftrightarrow \exists y \forall x (x = y \to φ)$