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Metamath Proof Explorer

Theorem 19.9dev

Description: 19.9d in the case of an existential quantifier, avoiding the ax-10 from nfex that would be used for the hypothesis of 19.9d , at the cost of an additional DV condition on y , ph . (Contributed by SN, 26-May-2024)

		Ref	Expression
	Hypothesis	19.9dev.1	$⊢ φ \to Ⅎ x ψ$
	Assertion	19.9dev	$⊢ φ \to (\exists x \exists y ψ \leftrightarrow \exists y ψ)$

Proof

Step	Hyp	Ref	Expression
1		19.9dev.1	$⊢ φ \to Ⅎ x ψ$
2		excom	$⊢ \exists x \exists y ψ \leftrightarrow \exists y \exists x ψ$
3		19.9t	$⊢ Ⅎ x ψ \to (\exists x ψ \leftrightarrow ψ)$
4	1 3	syl	$⊢ φ \to (\exists x ψ \leftrightarrow ψ)$
5	4	exbidv	$⊢ φ \to (\exists y \exists x ψ \leftrightarrow \exists y ψ)$
6	2 5	bitrid	$⊢ φ \to (\exists x \exists y ψ \leftrightarrow \exists y ψ)$