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Theorem sbcbi1 3827
Description: Distribution of class substitution over biconditional. One direction of sbcbig 3820 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcbi1 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcbi1
StepHypRef Expression
1 sbcex 3779 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 sbcbig 3820 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
32biimpd 230 . 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  wb 207  wcel 2105  Vcvv 3492  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494  df-sbc 3770
This theorem is referenced by:  dfconngr1  27894
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