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Theorem sbcim2gVD 42168
Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3745. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 41831 is sbcim2gVD 42168 without virtual deductions and was automatically derived from sbcim2gVD 42168.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓𝜒))   )
3:1,2: (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))   )
4:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
5:3,4: (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒))   )
6:5: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒)))   )
7:: (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
8:4,7: (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 [𝐴 / 𝑥](𝜓𝜒))   )
9:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓 𝜒)) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)))   )
10:8,9: (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓 𝜒))   )
11:10: (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 𝜒)))   )
12:6,11: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒)))   )
qed:12: (𝐴𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 [𝐴 / 𝑥]𝜒))))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcim2gVD (𝐴𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))

Proof of Theorem sbcim2gVD
StepHypRef Expression
1 idn1 41867 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 41906 . . . . . 6 (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓𝜒))   )
3 sbcimg 3745 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))))
43biimpd 232 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))))
51, 2, 4e12 42017 . . . . 5 (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓𝜒))   ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))   )
6 sbcimg 3745 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
71, 6e1a 41920 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
8 imbi2 352 . . . . . 6 (([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
98biimpcd 252 . . . . 5 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)) → (([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
105, 7, 9e21 42023 . . . 4 (   𝐴𝐵   ,   [𝐴 / 𝑥](𝜑 → (𝜓𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
1110in2 41898 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))   )
12 idn2 41906 . . . . . 6 (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   )
13 biimpr 223 . . . . . . 7 (([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
1413imim2d 57 . . . . . 6 (([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))))
157, 12, 14e12 42017 . . . . 5 (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))   )
161, 3e1a 41920 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)))   )
17 biimpr 223 . . . . . 6 (([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))) → (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓𝜒))))
1817com12 32 . . . . 5 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)) → (([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒))) → [𝐴 / 𝑥](𝜑 → (𝜓𝜒))))
1915, 16, 18e21 42023 . . . 4 (   𝐴𝐵   ,   ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))   ▶   [𝐴 / 𝑥](𝜑 → (𝜓𝜒))   )
2019in2 41898 . . 3 (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓𝜒)))   )
21 impbi 211 . . 3 (([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))) → ((([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓𝜒))) → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))))
2211, 20, 21e11 41981 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))   )
2322in1 41864 1 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2110  [wsbc 3694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3695  df-vd1 41863  df-vd2 41871
This theorem is referenced by: (None)
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