cbveuALT

Description: Alternative proof of cbveu . Since df-eu combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbveu.1	$⊢ Ⅎ y φ$
	cbveu.2	$⊢ Ⅎ x ψ$
	cbveu.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	cbveuALT	$⊢ \exists! x φ \leftrightarrow \exists! y ψ$

Proof

Step	Hyp	Ref	Expression
1		cbveu.1	$⊢ Ⅎ y φ$
2		cbveu.2	$⊢ Ⅎ x ψ$
3		cbveu.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
4	1 2 3	cbvex	$⊢ \exists x φ \leftrightarrow \exists y ψ$
5	1 2 3	cbvmo	$⊢ \exists^{} x φ \leftrightarrow \exists^{} y ψ$
6	4 5	anbi12i	$⊢ (\exists x φ \land \exists^{} x φ) \leftrightarrow (\exists y ψ \land \exists^{} y ψ)$
7		df-eu	$⊢ \exists! x φ \leftrightarrow (\exists x φ \land \exists^{*} x φ)$
8		df-eu	$⊢ \exists! y ψ \leftrightarrow (\exists y ψ \land \exists^{*} y ψ)$
9	6 7 8	3bitr4i	$⊢ \exists! x φ \leftrightarrow \exists! y ψ$

Metamath Proof Explorer

Theorem cbveuALT

Proof