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Theorem 2pm13.193VD 44579
Description: Virtual deduction proof of 2pm13.193 44228. 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 44228 is 2pm13.193VD 44579 without virtual deductions and was automatically derived from 2pm13.193VD 44579. (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 44250 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2 simpl 481 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
31, 2e1a 44303 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
4 simpl 481 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
53, 4e1a 44303 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
6 simpr 483 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
71, 6e1a 44303 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
8 pm3.21 470 . . . . . . . . . 10 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)))
95, 7, 8e11 44364 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
10 sbequ2 2237 . . . . . . . . . 10 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑣 / 𝑦]𝜑))
1110imdistanri 568 . . . . . . . . 9 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑣 / 𝑦]𝜑𝑥 = 𝑢))
129, 11e1a 44303 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
13 simpl 481 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑣 / 𝑦]𝜑)
1412, 13e1a 44303 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
15 simpr 483 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
163, 15e1a 44303 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
17 pm3.2 468 . . . . . . 7 ([𝑣 / 𝑦]𝜑 → (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)))
1814, 16, 17e11 44364 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
19 sbequ2 2237 . . . . . . 7 (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝜑))
2019imdistanri 568 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → (𝜑𝑦 = 𝑣))
2118, 20e1a 44303 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
22 simpl 481 . . . . 5 ((𝜑𝑦 = 𝑣) → 𝜑)
2321, 22e1a 44303 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝜑   )
24 pm3.2 468 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
253, 23, 24e11 44364 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
2625in1 44247 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
27 idn1 44250 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
28 simpl 481 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
2927, 28e1a 44303 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
3029, 4e1a 44303 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
3129, 15e1a 44303 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
32 simpr 483 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
3327, 32e1a 44303 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑   )
34 pm3.21 470 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → (𝜑𝑦 = 𝑣)))
3531, 33, 34e11 44364 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
36 sbequ1 2236 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
3736imdistanri 568 . . . . . . . . 9 ((𝜑𝑦 = 𝑣) → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣))
3835, 37e1a 44303 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
39 simpl 481 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → [𝑣 / 𝑦]𝜑)
4038, 39e1a 44303 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
41 pm3.21 470 . . . . . . 7 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)))
4230, 40, 41e11 44364 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
43 sbequ1 2236 . . . . . . 7 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
4443imdistanri 568 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢))
4542, 44e1a 44303 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
46 simpl 481 . . . . 5 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
4745, 46e1a 44303 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
48 pm3.2 468 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
4929, 47, 48e11 44364 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
5049in1 44247 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
5126, 50impbii 208 1 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  [wsb 2060
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 1906  ax-6 1964  ax-7 2004  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-sb 2061  df-vd1 44246
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator