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Metamath Proof Explorer

Theorem bj-equsvt

Description: A variant of equsv . (Contributed by BJ, 7-Oct-2024)

		Ref	Expression
	Assertion	bj-equsvt	$⊢ Ⅎ' x φ \to (\forall x (x = y \to φ) \leftrightarrow φ)$

Step	Hyp	Ref	Expression
1		bj-19.23t	$⊢ Ⅎ' x φ \to (\forall x (x = y \to φ) \leftrightarrow (\exists x x = y \to φ))$
2		ax6ev	$⊢ \exists x x = y$
3	2	a1bi	$⊢ φ \leftrightarrow (\exists x x = y \to φ)$
4	1 3	bitr4di	$⊢ Ⅎ' x φ \to (\forall x (x = y \to φ) \leftrightarrow φ)$