Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rspsbc2VD Structured version   Visualization version   GIF version

Theorem rspsbc2VD 44875
Description: Virtual deduction proof of rspsbc2 44554. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   𝐴𝐵   ,   𝐶𝐷   ▶   𝐶𝐷   )
3:: (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐵𝑦𝐷𝜑   )
4:1,3,?: e13 44768 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   [𝐴 / 𝑥]𝑦𝐷𝜑   )
5:1,4,?: e13 44768 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑦𝐷[𝐴 / 𝑥]𝜑   )
6:2,5,?: e23 44775 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
7:6: (   𝐴𝐵   ,   𝐶𝐷   ▶   (∀𝑥𝐵 𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
8:7: (   𝐴𝐵   ▶   (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
qed:8: (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rspsbc2VD (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem rspsbc2VD
StepHypRef Expression
1 idn2 44633 . . . . 5 (   𝐴𝐵   ,   𝐶𝐷   ▶   𝐶𝐷   )
2 idn1 44594 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn3 44635 . . . . . . 7 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐵𝑦𝐷 𝜑   )
4 rspsbc 3879 . . . . . . 7 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐴 / 𝑥]𝑦𝐷 𝜑))
52, 3, 4e13 44768 . . . . . 6 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   [𝐴 / 𝑥]𝑦𝐷 𝜑   )
6 sbcralg 3874 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 [𝐴 / 𝑥]𝜑))
76biimpd 229 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷 𝜑 → ∀𝑦𝐷 [𝐴 / 𝑥]𝜑))
82, 5, 7e13 44768 . . . . 5 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑦𝐷 [𝐴 / 𝑥]𝜑   )
9 rspsbc 3879 . . . . 5 (𝐶𝐷 → (∀𝑦𝐷 [𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
101, 8, 9e23 44775 . . . 4 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
1110in3 44629 . . 3 (   𝐴𝐵   ,   𝐶𝐷   ▶   (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
1211in2 44625 . 2 (   𝐴𝐵   ▶   (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
1312in1 44591 1 (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3061  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-v 3482  df-sbc 3789  df-vd1 44590  df-vd2 44598  df-vd3 44610
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator