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Theorem sbcim1 3834
Description: Distribution of class substitution over implication. One direction of sbcimg 3829 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 26-Oct-2024.)
Assertion
Ref Expression
sbcim1 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcim1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3788 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 dfsbcq2 3781 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
3 dfsbcq2 3781 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
4 dfsbcq2 3781 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
53, 4imbi12d 345 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
62, 5imbi12d 345 . . 3 (𝑦 = 𝐴 → (([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) ↔ ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))))
7 sbi1 2075 . . 3 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
86, 7vtoclg 3557 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
91, 8mpcom 38 1 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  [wsb 2068  wcel 2107  Vcvv 3475  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-sbc 3779
This theorem is referenced by:  sbcimdvOLD  3853  opsbc2ie  31716  frege59c  42673  frege60c  42674  frege62c  42676  frege65c  42679  frege70  42684  frege72  42686  frege92  42706  frege120  42734
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