Theorem sbimd

Description: Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022) Revise df-sb . (Revised by Steven Nguyen, 9-Jul-2023)

	Ref	Expression
Hypotheses	sbimd.1	⊢ Ⅎ 𝑥 𝜑
	sbimd.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion	sbimd	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 → [ 𝑦 / 𝑥 ] 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		sbimd.1	⊢ Ⅎ 𝑥 𝜑
2		sbimd.2	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3	1 2	alrimi	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) )
4		spsbim	⊢ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( [ 𝑦 / 𝑥 ] 𝜓 → [ 𝑦 / 𝑥 ] 𝜒 ) )
5	3 4	syl	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 → [ 𝑦 / 𝑥 ] 𝜒 ) )