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Metamath Proof Explorer

Theorem spsbim

Description: Distribute substitution over implication. Closed form of sbimi . Specialization of implication. (Contributed by NM, 5-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011) Revise df-sb . (Revised by BJ, 22-Dec-2020) (Proof shortened by Steven Nguyen, 24-Jul-2023)

		Ref	Expression
	Assertion	spsbim	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		stdpc4	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → [ 𝑡 / 𝑥 ] ( 𝜑 → 𝜓 ) )
2		sbi1	⊢ ( [ 𝑡 / 𝑥 ] ( 𝜑 → 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 ) )
3	1 2	syl	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝑡 / 𝑥 ] 𝜑 → [ 𝑡 / 𝑥 ] 𝜓 ) )