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Theorem sbim 2287
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 2284 . 2 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbi2 2285 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
31, 2impbii 197 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  [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:  sbrim  2288  sblim  2289  sbor  2290  sban  2291  sbbi  2293  sbequ8ALT  2299  sbcimg  3348  mo5f  28497  iuninc  28550  suppss2f  28608  esumpfinvalf  29262  bj-sbnf  31858  wl-sbrimt  32385  wl-sblimt  32386  frege58bcor  37099  frege60b  37101  frege65b  37106  ellimcabssub0  38570
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