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Theorem sbcan 3630
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 3597 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 sbcex 3597 . . 3 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
32adantl 467 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → 𝐴 ∈ V)
4 dfsbcq2 3590 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
5 dfsbcq2 3590 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
6 dfsbcq2 3590 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
75, 6anbi12d 616 . . 3 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
8 sban 2546 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
94, 7, 8vtoclbg 3418 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
101, 3, 9pm5.21nii 367 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  [wsb 2049  wcel 2145  Vcvv 3351  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  sbc3an  3645  sbcabel  3666  csbopg  4557  csbuni  4602  csbmpt12  5143  csbxp  5340  difopab  5392  sbcfung  6055  sbcfng  6182  sbcfg  6183  fmptsnd  6579  f1od2  29839  esum2dlem  30494  bnj976  31186  bnj110  31266  bnj1040  31378  csbwrecsg  33510  csboprabg  33513  csbmpt22g  33514  f1omptsnlem  33520  mptsnunlem  33522  relowlpssretop  33549  csbfinxpg  33562  sbcani  34242  sbccom2lem  34261  brtrclfv2  38545  cotrclrcl  38560  frege124d  38579  sbiota1  39161  onfrALTlem5  39282  onfrALTlem4  39283  csbxpgOLD  39576  onfrALTlem5VD  39643
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