Theorem bj-equsvt

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

	Ref	Expression
Assertion	bj-equsvt	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-19.23t	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜑 ) ) )
2		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
3	2	a1bi	⊢ ( 𝜑 ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜑 ) )
4	1 3	bitr4di	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜑 ) )