Theorem 2pm13.193VD 44899

Description: Virtual deduction proof of 2pm13.193 44549. 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 44549 is 2pm13.193VD 44899 without virtual deductions and was automatically derived from 2pm13.193VD 44899. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

1::	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2:1:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
3:2:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
4:1:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
5:3,4:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
6:5:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
7:6:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
8:2:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
9:7,8:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   )
10:9:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   (𝜑 ∧ 𝑦 = 𝑣)   )
11:10:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   𝜑   )
12:2,11:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][ 𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
13:12:	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
14::	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
15:14:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
16:15:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
17:14:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑    )
18:16,17:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ( 𝜑 ∧ 𝑦 = 𝑣)   )
19:18:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   )
20:15:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
21:19:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
22:20,21:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
23:22:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([ 𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
24:23:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
25:15,24:	⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
26:25:	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
qed:13,26:	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Assertion

Ref	Expression
2pm13.193VD	⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Proof of Theorem 2pm13.193VD

Step	Hyp	Ref	Expression
1		idn1 44571	. . . . 5 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
2		simpl 482	. . . . 5 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
3	1, 2	e1a 44624	. . . 4 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
4		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢)
5	3, 4	e1a 44624	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑥 = 𝑢   )
6		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
7	1, 6	e1a 44624	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
8		pm3.21 471	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)))
9	5, 7, 8	e11 44685	. . . . . . . . 9 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
10		sbequ2 2250	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → [𝑣 / 𝑦]𝜑))
11	10	imdistanri 569	. . . . . . . . 9 ⊢ (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) → ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢))
12	9, 11	e1a 44624	. . . . . . . 8 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
13		simpl 482	. . . . . . . 8 ⊢ (([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) → [𝑣 / 𝑦]𝜑)
14	12, 13	e1a 44624	. . . . . . 7 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
15		simpr 484	. . . . . . . 8 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣)
16	3, 15	e1a 44624	. . . . . . 7 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝑦 = 𝑣   )
17		pm3.2 469	. . . . . . 7 ⊢ ([𝑣 / 𝑦]𝜑 → (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)))
18	14, 16, 17	e11 44685	. . . . . 6 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   )
19		sbequ2 2250	. . . . . . 7 ⊢ (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑 → 𝜑))
20	19	imdistanri 569	. . . . . 6 ⊢ (([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣) → (𝜑 ∧ 𝑦 = 𝑣))
21	18, 20	e1a 44624	. . . . 5 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   (𝜑 ∧ 𝑦 = 𝑣)   )
22		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑣) → 𝜑)
23	21, 22	e1a 44624	. . . 4 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   𝜑   )
24		pm3.2 469	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝜑 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)))
25	3, 23, 24	e11 44685	. . 3 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
26	25	in1 44568	. 2 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
27		idn1 44571	. . . . 5 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
28		simpl 482	. . . . 5 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
29	27, 28	e1a 44624	. . . 4 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
30	29, 4	e1a 44624	. . . . . . 7 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
31	29, 15	e1a 44624	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
32		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
33	27, 32	e1a 44624	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑   )
34		pm3.21 471	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝜑 → (𝜑 ∧ 𝑦 = 𝑣)))
35	31, 33, 34	e11 44685	. . . . . . . . 9 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝜑 ∧ 𝑦 = 𝑣)   )
36		sbequ1 2249	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
37	36	imdistanri 569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝑣) → ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣))
38	35, 37	e1a 44624	. . . . . . . 8 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   )
39		simpl 482	. . . . . . . 8 ⊢ (([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣) → [𝑣 / 𝑦]𝜑)
40	38, 39	e1a 44624	. . . . . . 7 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑣 / 𝑦]𝜑   )
41		pm3.21 471	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)))
42	30, 40, 41	e11 44685	. . . . . 6 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
43		sbequ1 2249	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ([𝑣 / 𝑦]𝜑 → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
44	43	imdistanri 569	. . . . . 6 ⊢ (([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢))
45	42, 44	e1a 44624	. . . . 5 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
46		simpl 482	. . . . 5 ⊢ (([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢) → [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
47	45, 46	e1a 44624	. . . 4 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   [𝑢 / 𝑥][𝑣 / 𝑦]𝜑   )
48		pm3.2 469	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)))
49	29, 47, 48	e11 44685	. . 3 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
50	49	in1 44568	. 2 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑))
51	26, 50	impbii 209	1 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  [wsb 2065

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 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-vd1 44567

This theorem is referenced by: (None)