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Theorem sbi2v 2278
 Description: Move implication into substitution. Version of sbi2 2468 with a disjoint variable condition, not requiring ax-13 2333. (Contributed by Wolf Lammen, 18-Jan-2023.)
Assertion
Ref Expression
sbi2v (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbi2v
StepHypRef Expression
1 sbnv 2276 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2 pm2.21 121 . . . 4 𝜑 → (𝜑𝜓))
32sbimi 2017 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑𝜓))
41, 3sylbir 227 . 2 (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑𝜓))
5 ax-1 6 . . 3 (𝜓 → (𝜑𝜓))
65sbimi 2017 . 2 ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓))
74, 6ja 175 1 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  [wsb 2011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012 This theorem is referenced by:  sbimv  2279
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