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Theorem sbi2 2285
Description: Introduction of implication into substitution. (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 2283 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2 pm2.21 118 . . . 4 𝜑 → (𝜑𝜓))
32sbimi 1836 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑𝜓))
41, 3sylbir 223 . 2 (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑𝜓))
5 ax-1 6 . . 3 (𝜓 → (𝜑𝜓))
65sbimi 1836 . 2 ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓))
74, 6ja 171 1 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  [wsb 1830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1699  df-sb 1831
This theorem is referenced by:  sbim  2287
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