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Theorem sbim 2471
Description: Implication inside and outside of a substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbimv 2280. (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 2468 . 2 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbi2 2469 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
31, 2impbii 201 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  [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:  sbrim  2472  sblim  2473  sbor  2474  sban  2475  sbbi  2477  sbequ8ALT  2483  sbcimg  3695  mo5f  29900  iuninc  29945  suppss2f  30008  esumpfinvalf  30740  bj-sbnf  33407  wl-sbrimt  33929  wl-sblimt  33930  frege58bcor  39163  frege60b  39165  frege65b  39170  ellimcabssub0  40767
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