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Theorem 2pm13.193VD 38661
Description: Virtual deduction proof of 2pm13.193 38289. 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 38289 is 2pm13.193VD 38661 without virtual deductions and was automatically derived from 2pm13.193VD 38661. (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 38311 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2 simpl 473 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
31, 2e1a 38373 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
4 simpl 473 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
53, 4e1a 38373 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
6 simpr 477 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
71, 6e1a 38373 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
8 pm3.21 464 . . . . . . . . . 10 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)))
95, 7, 8e11 38434 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
10 sbequ2 1879 . . . . . . . . . 10 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑣 / 𝑦]𝜑))
1110imdistanri 726 . . . . . . . . 9 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑣 / 𝑦]𝜑𝑥 = 𝑢))
129, 11e1a 38373 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
13 simpl 473 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑣 / 𝑦]𝜑)
1412, 13e1a 38373 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
15 simpr 477 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
163, 15e1a 38373 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
17 pm3.2 463 . . . . . . 7 ([𝑣 / 𝑦]𝜑 → (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)))
1814, 16, 17e11 38434 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
19 sbequ2 1879 . . . . . . 7 (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝜑))
2019imdistanri 726 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → (𝜑𝑦 = 𝑣))
2118, 20e1a 38373 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
22 simpl 473 . . . . 5 ((𝜑𝑦 = 𝑣) → 𝜑)
2321, 22e1a 38373 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝜑   )
24 pm3.2 463 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
253, 23, 24e11 38434 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
2625in1 38308 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
27 idn1 38311 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
28 simpl 473 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
2927, 28e1a 38373 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
3029, 4e1a 38373 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
3129, 15e1a 38373 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
32 simpr 477 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
3327, 32e1a 38373 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑   )
34 pm3.21 464 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → (𝜑𝑦 = 𝑣)))
3531, 33, 34e11 38434 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
36 sbequ1 2107 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
3736imdistanri 726 . . . . . . . . 9 ((𝜑𝑦 = 𝑣) → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣))
3835, 37e1a 38373 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
39 simpl 473 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → [𝑣 / 𝑦]𝜑)
4038, 39e1a 38373 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
41 pm3.21 464 . . . . . . 7 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)))
4230, 40, 41e11 38434 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
43 sbequ1 2107 . . . . . . 7 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
4443imdistanri 726 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢))
4542, 44e1a 38373 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
46 simpl 473 . . . . 5 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
4745, 46e1a 38373 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
48 pm3.2 463 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
4929, 47, 48e11 38434 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
5049in1 38308 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
5126, 50impbii 199 1 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878  df-vd1 38307
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator