Theorem bj-equsalvwd

Description: Variant of equsalvw . (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Hypotheses	bj-equsalvwd.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-equsalvwd.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
	bj-equsalvwd.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	bj-equsalvwd	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-equsalvwd.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		bj-equsalvwd.nf	⊢ ( 𝜑 → Ⅎ' 𝑥 𝜒 )
3		bj-equsalvwd.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
4	3	pm5.74da	⊢ ( 𝜑 → ( ( 𝑥 = 𝑦 → 𝜓 ) ↔ ( 𝑥 = 𝑦 → 𝜒 ) ) )
5	1 4	albidh	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜒 ) ) )
6		bj-equsvt	⊢ ( Ⅎ' 𝑥 𝜒 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜒 ) ↔ 𝜒 ) )
7	2 6	syl	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜒 ) ↔ 𝜒 ) )
8	5 7	bitrd	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ↔ 𝜒 ) )