Theorem spsbim 2050

Description: Distribute substitution over implication. Closed form of sbimi 2052. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2043. (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 2046	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑡 / 𝑥](𝜑 → 𝜓))
2		sbi1 2049	. 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 1520  [wsb 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888

This theorem depends on definitions:  df-bi 208  df-sb 2043

This theorem is referenced by:  spsbbi  2051  sbimdv  2056  sbimd  2209  mo3  2604  mo3OLD  2605  bj-hbsb3t  33654  wl-mo3t  34343  pm11.59  40261  sbiota1  40304