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Theorem sbcim1 3791
Description: Distribution of class substitution over implication. One direction of sbcimg 3786 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2146, ax-12 2182. (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 3747 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 dfsbcq2 3740 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
3 dfsbcq2 3740 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
4 dfsbcq2 3740 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
53, 4imbi12d 344 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
62, 5imbi12d 344 . . 3 (𝑦 = 𝐴 → (([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) ↔ ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))))
7 sbi1 2076 . . 3 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
86, 7vtoclg 3508 . 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 1541  [wsb 2067  wcel 2113  Vcvv 3437  [wsbc 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-sbc 3738
This theorem is referenced by:  opsbc2ie  32459  frege59c  44042  frege60c  44043  frege62c  44045  frege65c  44048  frege70  44053  frege72  44055  frege92  44075  frege120  44103
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