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Theorem sbcim1 3823
Description: Distribution of class substitution over implication. One direction of sbcimg 3818 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcim1 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcim1
StepHypRef Expression
1 sbcex 3780 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 sbcimg 3818 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
32biimpd 231 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
41, 3mpcom 38 1 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3493  [wsbc 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-v 3495  df-sbc 3771
This theorem is referenced by:  sbcimdv  3841  opsbc2ie  30231  frege59c  40253  frege60c  40254  frege62c  40256  frege65c  40259  frege70  40264  frege72  40266  frege92  40286  frege120  40314
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