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Theorem sbcimg 3675
Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
sbcimg (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcimg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3636 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
2 dfsbcq2 3636 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3 dfsbcq2 3636 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
42, 3imbi12d 335 . 2 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
5 sbim 2554 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
61, 4, 5vtoclbg 3460 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  [wsb 2060  wcel 2156  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393  df-sbc 3634
This theorem is referenced by:  sbcim1  3680  sbceqal  3685  sbc19.21g  3698  sbcssg  4278  iota4an  6079  sbcfung  6121  riotass2  6858  tfinds2  7289  telgsums  18588  bnj110  31246  bnj92  31250  bnj539  31279  bnj540  31280  f1omptsnlem  33495  mptsnunlem  33497  topdifinffinlem  33506  relowlpssretop  33523  rdgeqoa  33529  sbcimi  34218  cdlemkid3N  36708  cdlemkid4  36709  cdlemk35s  36712  cdlemk39s  36714  cdlemk42  36716  frege77  38728  frege116  38767  frege118  38769  sbcim2g  39240  sbcssOLD  39248  onfrALTlem5  39249  sbcim2gVD  39599  sbcssgVD  39607  onfrALTlem5VD  39609  iccelpart  41938
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