This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sbimi

Description: Distribute substitution over implication. (Contributed by NM, 25-Jun-1998) Revise df-sb . (Revised by BJ, 22-Dec-2020) (Proof shortened by Steven Nguyen, 24-Jul-2023)

		Ref	Expression
	Hypothesis	sbimi.1	⊢ ( 𝜑 → 𝜓 )
	Assertion	sbimi	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbimi.1	⊢ ( 𝜑 → 𝜓 )
2	1	sbt	⊢ [ 𝑡 / 𝑥 ] ( 𝜑 → 𝜓 )
3		sbi1	⊢ ( [ 𝑡 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 ) )
4	2 3	ax-mp	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 )