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Theorem sbcan 3676
Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcan ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcan
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3643 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 sbcex 3643 . . 3 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
32adantl 469 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → 𝐴 ∈ V)
4 dfsbcq2 3636 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
5 dfsbcq2 3636 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
6 dfsbcq2 3636 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
75, 6anbi12d 618 . . 3 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
8 sban 2558 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
94, 7, 8vtoclbg 3460 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
101, 3, 9pm5.21nii 369 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  [wsb 2060  wcel 2156  Vcvv 3391  [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:  sbc3an  3691  sbcabel  3712  csbopg  4613  csbuni  4660  csbmpt12  5205  csbxp  5402  difopab  5455  sbcfung  6125  sbcfng  6253  sbcfg  6254  fmptsnd  6660  f1od2  29826  esum2dlem  30479  bnj976  31171  bnj110  31251  bnj1040  31363  csbwrecsg  33490  csboprabg  33493  csbmpt22g  33494  f1omptsnlem  33500  mptsnunlem  33502  relowlpssretop  33528  csbfinxpg  33541  sbcani  34221  sbccom2lem  34239  brtrclfv2  38519  cotrclrcl  38534  frege124d  38553  sbiota1  39134  onfrALTlem5  39255  onfrALTlem4  39256  csbxpgOLD  39548  onfrALTlem5VD  39615
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