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Theorem sbi2 2276
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 2274 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2 pm2.21 113 . . . 4 𝜑 → (𝜑𝜓))
32sbimi 1834 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑𝜓))
41, 3sylbir 220 . 2 (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑𝜓))
5 ax-1 6 . . 3 (𝜓 → (𝜑𝜓))
65sbimi 1834 . 2 ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓))
74, 6ja 168 1 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  [wsb 1828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829
This theorem is referenced by:  sbim  2278
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