Theorem spsbim 2072

Description: Distribute substitution over implication. Closed form of sbimi 2074. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2065. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)

Assertion

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

Proof of Theorem spsbim

Step	Hyp	Ref	Expression
1		stdpc4 2068	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑡 / 𝑥](𝜑 → 𝜓))
2		sbi1 2071	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-sb 2065

This theorem is referenced by:  spsbbi  2073  sbimdv  2078  sbimd  2246  mo3  2567  bj-hbsb3t  36754  wl-mo3t  37530  ss2ab1  42212  pm11.59  44360  sbiota1  44403