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Theorem sbim 2278
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 2275 . 2 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbi2 2276 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
31, 2impbii 194 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  [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:  sbrim  2279  sblim  2280  sbor  2281  sban  2282  sbbi  2284  sbequ8ALT  2290  sbcimg  3333  mo5f  28281  iuninc  28335  suppss2fOLD  28395  suppss2f  28396  esumpfinvalf  29052  bj-sbnf  31625  wl-sbrimt  32109  wl-sblimt  32110  frege58bcor  36738  frege60b  36740  frege65b  36745  ellimcabssub0  38101
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