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Theorem sbimdv 2030
Description: Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2173. (Contributed by Wolf Lammen, 6-May-2023.) Revise df-sb 2017. (Revised by Steven Nguyen, 6-Jul-2023.)
Hypothesis
Ref Expression
sbimdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbimdv (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑡)   𝜓(𝑥,𝑡)   𝜒(𝑥,𝑡)

Proof of Theorem sbimdv
StepHypRef Expression
1 sbimdv.1 . . 3 (𝜑 → (𝜓𝜒))
21alrimiv 1887 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 spsbim 2024 . 2 (∀𝑥(𝜓𝜒) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
42, 3syl 17 1 (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1506  [wsb 2016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870
This theorem depends on definitions:  df-bi 199  df-sb 2017
This theorem is referenced by:  sbbidvOLD  2438
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