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Theorem sbim 2399
Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sbim ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbim
StepHypRef Expression
1 sbi1 2396 . 2 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbi2 2397 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
31, 2impbii 199 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1883
This theorem is referenced by:  sbrim  2400  sblim  2401  sbor  2402  sban  2403  sbbi  2405  sbequ8ALT  2411  sbcimg  3464  mo5f  29164  iuninc  29215  suppss2f  29272  esumpfinvalf  29911  bj-sbnf  32444  wl-sbrimt  32939  wl-sblimt  32940  frege58bcor  37646  frege60b  37648  frege65b  37653  ellimcabssub0  39221
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