Theorem bj-sbievw

Description: Lemma for substitution. Closed form of equsalvw and sbievw . (Contributed by BJ, 23-Jul-2023)

	Ref	Expression
Assertion	bj-sbievw	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sb6	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) )
2		bj-sblem	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜓 ) ) )
3		sb6	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
4		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
5	4	a1bi	⊢ ( 𝜓 ↔ ( ∃ 𝑥 𝑥 = 𝑦 → 𝜓 ) )
6	2 3 5	3bitr4g	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
7	1 6	sylbi	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) )