Theorem sbim

Description: Implication inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993)

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

Proof

Step	Hyp	Ref	Expression
1		sbi1	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )
2		sbi2	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) → [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) )
3	1 2	impbii	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜓 ) )