Theorem sbrimvwOLD

Description: Obsolete version of sbrimvw as of 5-Jun-2026. (Contributed by Wolf Lammen, 29-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sbrimvwOLD	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		sb6	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) )
2		bi2.04	⊢ ( ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) ) ↔ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) )
3	2	albii	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) )
4		19.21v	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 = 𝑦 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) )
5	1 3 4	3bitr2i	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) )
6		sb6	⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) )
7	6	imbi2i	⊢ ( ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜓 ) ) )
8	5 7	bitr4i	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )