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Theorem spsbim 2077
Description: Distribute substitution over implication. Closed form of sbimi 2079. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)
Assertion
Ref Expression
spsbim (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Proof of Theorem spsbim
StepHypRef Expression
1 stdpc4 2073 . 2 (∀𝑥(𝜑𝜓) → [𝑡 / 𝑥](𝜑𝜓))
2 sbi1 2076 . 2 ([𝑡 / 𝑥](𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
31, 2syl 17 1 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-sb 2070
This theorem is referenced by:  spsbbi  2078  sbimdv  2083  sbimd  2244  mo3  2626  bj-hbsb3t  34220  wl-mo3t  34970  pm11.59  41082  sbiota1  41125
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