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Theorem sbim 2423
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 2420 . 2 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbi2 2421 . 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 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  sbrim  2424  sblim  2425  sbor  2426  sban  2427  sbbi  2429  sbequ8ALT  2435  sbcimg  3510  mo5f  29452  iuninc  29505  suppss2f  29567  esumpfinvalf  30266  bj-sbnf  32953  wl-sbrimt  33461  wl-sblimt  33462  frege58bcor  38514  frege60b  38516  frege65b  38521  ellimcabssub0  40167
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