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Theorem sbimd 2173
Description: Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2017. (Revised by Steven Nguyen, 9-Jul-2023.)
Hypotheses
Ref Expression
sbimd.1 𝑥𝜑
sbimd.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbimd (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))

Proof of Theorem sbimd
StepHypRef Expression
1 sbimd.1 . . 3 𝑥𝜑
2 sbimd.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2144 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbim 2024 . 2 (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
53, 4syl 17 1 (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1506  wnf 1747  [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  ax-6 1929  ax-7 1966  ax-12 2107
This theorem depends on definitions:  df-bi 199  df-ex 1744  df-nf 1748  df-sb 2017
This theorem is referenced by:  sbbidOLD  2176  spsbimvOLD  2252
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