Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2pm13.193VD Structured version   Visualization version   GIF version

Theorem 2pm13.193VD 41228
Description: Virtual deduction proof of 2pm13.193 40877. 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. 2pm13.193 40877 is 2pm13.193VD 41228 without virtual deductions and was automatically derived from 2pm13.193VD 41228. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2:1: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
3:2: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
4:1: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
5:3,4: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
6:5: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
7:6: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
8:2: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
9:7,8: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
10:9: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
11:10: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝜑   )
12:2,11: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
13:12: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
14:: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
15:14: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
16:15: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
17:14: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑    )
18:16,17: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ( 𝜑𝑦 = 𝑣)   )
19:18: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
20:15: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
21:19: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
22:20,21: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
23:22: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
24:23: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
25:15,24: (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
26:25: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
qed:13,26: (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Assertion
Ref Expression
2pm13.193VD (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))

Proof of Theorem 2pm13.193VD
StepHypRef Expression
1 idn1 40899 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2 simpl 485 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
31, 2e1a 40952 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
4 simpl 485 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
53, 4e1a 40952 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
6 simpr 487 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
71, 6e1a 40952 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
8 pm3.21 474 . . . . . . . . . 10 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)))
95, 7, 8e11 41013 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
10 sbequ2 2243 . . . . . . . . . 10 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑣 / 𝑦]𝜑))
1110imdistanri 572 . . . . . . . . 9 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑣 / 𝑦]𝜑𝑥 = 𝑢))
129, 11e1a 40952 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
13 simpl 485 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑣 / 𝑦]𝜑)
1412, 13e1a 40952 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
15 simpr 487 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
163, 15e1a 40952 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
17 pm3.2 472 . . . . . . 7 ([𝑣 / 𝑦]𝜑 → (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)))
1814, 16, 17e11 41013 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
19 sbequ2 2243 . . . . . . 7 (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝜑))
2019imdistanri 572 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → (𝜑𝑦 = 𝑣))
2118, 20e1a 40952 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
22 simpl 485 . . . . 5 ((𝜑𝑦 = 𝑣) → 𝜑)
2321, 22e1a 40952 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝜑   )
24 pm3.2 472 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
253, 23, 24e11 41013 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
2625in1 40896 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
27 idn1 40899 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
28 simpl 485 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
2927, 28e1a 40952 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
3029, 4e1a 40952 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
3129, 15e1a 40952 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
32 simpr 487 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
3327, 32e1a 40952 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑   )
34 pm3.21 474 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → (𝜑𝑦 = 𝑣)))
3531, 33, 34e11 41013 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
36 sbequ1 2242 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
3736imdistanri 572 . . . . . . . . 9 ((𝜑𝑦 = 𝑣) → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣))
3835, 37e1a 40952 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
39 simpl 485 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → [𝑣 / 𝑦]𝜑)
4038, 39e1a 40952 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
41 pm3.21 474 . . . . . . 7 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)))
4230, 40, 41e11 41013 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
43 sbequ1 2242 . . . . . . 7 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
4443imdistanri 572 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢))
4542, 44e1a 40952 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
46 simpl 485 . . . . 5 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
4745, 46e1a 40952 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
48 pm3.2 472 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
4929, 47, 48e11 41013 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
5049in1 40896 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
5126, 50impbii 211 1 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1531  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-12 2170
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-sb 2064  df-vd1 40895
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator