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Theorem sbi2 2469
 Description: Introduction of implication into substitution. For a version requiring disjoint variables, but fewer axioms, see sbi2v 2279. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sbi2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))

Proof of Theorem sbi2
StepHypRef Expression
1 sbn 2467 . . 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 2055  ax-10 2135  ax-12 2163  ax-13 2334 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:  sbim  2471
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