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Theorem 2pm13.193VD 39888
Description: Virtual deduction proof of 2pm13.193 39527. 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 39527 is 2pm13.193VD 39888 without virtual deductions and was automatically derived from 2pm13.193VD 39888. (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 39549 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2 simpl 475 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
31, 2e1a 39611 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
4 simpl 475 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
53, 4e1a 39611 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
6 simpr 478 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
71, 6e1a 39611 . . . . . . . . . 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 39672 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
10 sbequ2 2066 . . . . . . . . . 10 (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑣 / 𝑦]𝜑))
1110imdistanri 566 . . . . . . . . 9 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑣 / 𝑦]𝜑𝑥 = 𝑢))
129, 11e1a 39611 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
13 simpl 475 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑣 / 𝑦]𝜑)
1412, 13e1a 39611 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
15 simpr 478 . . . . . . . 8 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
163, 15e1a 39611 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
17 pm3.2 462 . . . . . . 7 ([𝑣 / 𝑦]𝜑 → (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)))
1814, 16, 17e11 39672 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
19 sbequ2 2066 . . . . . . 7 (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑𝜑))
2019imdistanri 566 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → (𝜑𝑦 = 𝑣))
2118, 20e1a 39611 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
22 simpl 475 . . . . 5 ((𝜑𝑦 = 𝑣) → 𝜑)
2321, 22e1a 39611 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝜑   )
24 pm3.2 462 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))
253, 23, 24e11 39672 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
2625in1 39546 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
27 idn1 39549 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   )
28 simpl 475 . . . . 5 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢𝑦 = 𝑣))
2927, 28e1a 39611 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢𝑦 = 𝑣)   )
3029, 4e1a 39611 . . . . . . 7 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
3129, 15e1a 39611 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
32 simpr 478 . . . . . . . . . . 11 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
3327, 32e1a 39611 . . . . . . . . . 10 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑   )
34 pm3.21 464 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → (𝜑𝑦 = 𝑣)))
3531, 33, 34e11 39672 . . . . . . . . 9 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝜑𝑦 = 𝑣)   )
36 sbequ1 2277 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
3736imdistanri 566 . . . . . . . . 9 ((𝜑𝑦 = 𝑣) → ([𝑣 / 𝑦]𝜑𝑦 = 𝑣))
3835, 37e1a 39611 . . . . . . . 8 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑦 = 𝑣)   )
39 simpl 475 . . . . . . . 8 (([𝑣 / 𝑦]𝜑𝑦 = 𝑣) → [𝑣 / 𝑦]𝜑)
4038, 39e1a 39611 . . . . . . 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 39672 . . . . . 6 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
43 sbequ1 2277 . . . . . . 7 (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
4443imdistanri 566 . . . . . 6 (([𝑣 / 𝑦]𝜑𝑥 = 𝑢) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢))
4542, 44e1a 39611 . . . . 5 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢)   )
46 simpl 475 . . . . 5 (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑𝑥 = 𝑢) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
4745, 46e1a 39611 . . . 4 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
48 pm3.2 462 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
4929, 47, 48e11 39672 . . 3 (   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
5049in1 39546 . 2 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
5126, 50impbii 201 1 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-sb 2065  df-vd1 39545
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator