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Theorem sb2imi 2073
Description: Distribute substitution over implication. Compare al2imi 1809. (Contributed by Steven Nguyen, 13-Aug-2023.)
Hypothesis
Ref Expression
sb2imi.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sb2imi ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))

Proof of Theorem sb2imi
StepHypRef Expression
1 sb2imi.1 . . 3 (𝜑 → (𝜓𝜒))
21sbimi 2072 . 2 ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥](𝜓𝜒))
3 sbi1 2069 . 2 ([𝑡 / 𝑥](𝜓𝜒) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
42, 3syl 17 1 ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-sb 2063
This theorem is referenced by:  sban  2079  sbn1  38968
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