Theorem bj-sbievw2

Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023)

	Ref	Expression
Assertion	bj-sbievw2	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜓 → 𝜑 ) → ( 𝜓 → [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sb6	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜓 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) ) )
2		bj-sblem2	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) ) → ( ( ∃ 𝑥 𝑥 = 𝑦 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3		jarr	⊢ ( ( ( ∃ 𝑥 𝑥 = 𝑦 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
4		sb6	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
5	3 4	imbitrrdi	⊢ ( ( ( ∃ 𝑥 𝑥 = 𝑦 → 𝜓 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ( 𝜓 → [ 𝑦 / 𝑥 ] 𝜑 ) )
6	2 5	syl	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → [ 𝑦 / 𝑥 ] 𝜑 ) )
7	1 6	sylbi	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜓 → 𝜑 ) → ( 𝜓 → [ 𝑦 / 𝑥 ] 𝜑 ) )