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Theorem sbcangOLD 38565
Description: Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcan 3476 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcangOLD (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcangOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3436 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
2 dfsbcq2 3436 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3 dfsbcq2 3436 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
42, 3anbi12d 747 . 2 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
5 sban 2398 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
61, 4, 5vtoclbg 3265 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  [wsb 1879  wcel 1989  [wsbc 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3200  df-sbc 3434
This theorem is referenced by:  csbunigOLD  38877  csbxpgOLD  38879  csbingVD  38946  onfrALTlem5VD  38947  onfrALTlem4VD  38948  csbxpgVD  38956  csbunigVD  38960
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