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Metamath Proof Explorer

Theorem 19.8aw

Description: If a formula is true, then it is true for at least one instance. This is to 19.8a what spw is to sp . (Contributed by SN, 26-Sep-2024)

		Ref	Expression
	Hypothesis	19.8aw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	19.8aw	$⊢ φ \to \exists x φ$

Proof

Step	Hyp	Ref	Expression
1		19.8aw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2		alnex	$⊢ \forall x \neg φ \leftrightarrow \neg \exists x φ$
3	1	notbid	$⊢ x = y \to (\neg φ \leftrightarrow \neg ψ)$
4	3	spw	$⊢ \forall x \neg φ \to \neg φ$
5	2 4	sylbir	$⊢ \neg \exists x φ \to \neg φ$
6	5	con4i	$⊢ φ \to \exists x φ$