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Theorem sbcim2g 43289
Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3828. sbcim2g 43289 is sbcim2gVD 43626 without virtual deductions and was automatically derived from sbcim2gVD 43626 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcim2g (𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))

Proof of Theorem sbcim2g
StepHypRef Expression
1 sbcimg 3828 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))))
21biimpd 228 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))))
3 sbcimg 3828 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
4 imbi2 348 . . . 4 (([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
54biimpcd 248 . . 3 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)) → (([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
62, 3, 5syl6ci 71 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
7 idd 24 . . . 4 (𝐴𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
8 biimpr 219 . . . 4 (([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
93, 7, 8ee13 43255 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))))
109, 1sylibrd 258 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓𝜒))))
116, 10impbid 211 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-sbc 3778
This theorem is referenced by:  trsbc  43291  trsbcVD  43628
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