Theorem 2pm13.193VD 44874

Description: Virtual deduction proof of 2pm13.193 44523. 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 44523 is 2pm13.193VD 44874 without virtual deductions and was automatically derived from 2pm13.193VD 44874. (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 44545	. . . . 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 44598	. . . 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 44598	. . . . . . . . . 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 44598	. . . . . . . . . 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 44659	. . . . . . . . 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 44598	. . . . . . . 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 44598	. . . . . . 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 44598	. . . . . . 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 44659	. . . . . 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 44598	. . . . 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 44598	. . . 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 44659	. . 3 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
26	25	in1 44542	. 2 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
27		idn1 44545	. . . . 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 44598	. . . 4 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
30	29, 4	e1a 44598	. . . . . . 7 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
31	29, 15	e1a 44598	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
32		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
33	27, 32	e1a 44598	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑   )
34		pm3.21 471	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝜑 → (𝜑 ∧ 𝑦 = 𝑣)))
35	31, 33, 34	e11 44659	. . . . . . . . 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 44598	. . . . . . . 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 44598	. . . . . . 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 44659	. . . . . 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 44598	. . . . 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 44598	. . . 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 44659	. . 3 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
50	49	in1 44542	. 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 1537  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-vd1 44541

This theorem is referenced by: (None)