Theorem 2pm13.193VD 45258

Description: Virtual deduction proof of 2pm13.193 44908. 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 44908 is 2pm13.193VD 45258 without virtual deductions and was automatically derived from 2pm13.193VD 45258. (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 44930	. . . . 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 44983	. . . 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 44983	. . . . . . . . . 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 44983	. . . . . . . . . 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 45044	. . . . . . . . 9 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
10		sbequ2 2257	. . . . . . . . . 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 44983	. . . . . . . 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 44983	. . . . . . 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 44983	. . . . . . 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 45044	. . . . . 6 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣)   )
19		sbequ2 2257	. . . . . . 7 ⊢ (𝑦 = 𝑣 → ([𝑣 / 𝑦]𝜑 → 𝜑))
20	19	imdistanri 569	. . . . . 6 ⊢ (([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣) → (𝜑 ∧ 𝑦 = 𝑣))
21	18, 20	e1a 44983	. . . . 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 44983	. . . 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 45044	. . 3 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   )
26	25	in1 44927	. 2 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
27		idn1 44930	. . . . 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 44983	. . . 4 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝑥 = 𝑢 ∧ 𝑦 = 𝑣)   )
30	29, 4	e1a 44983	. . . . . . 7 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑥 = 𝑢   )
31	29, 15	e1a 44983	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝑦 = 𝑣   )
32		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) → 𝜑)
33	27, 32	e1a 44983	. . . . . . . . . 10 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   𝜑   )
34		pm3.21 471	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝜑 → (𝜑 ∧ 𝑦 = 𝑣)))
35	31, 33, 34	e11 45044	. . . . . . . . 9 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   (𝜑 ∧ 𝑦 = 𝑣)   )
36		sbequ1 2256	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝜑 → [𝑣 / 𝑦]𝜑))
37	36	imdistanri 569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝑣) → ([𝑣 / 𝑦]𝜑 ∧ 𝑦 = 𝑣))
38	35, 37	e1a 44983	. . . . . . . 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 44983	. . . . . . 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 45044	. . . . . 6 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ([𝑣 / 𝑦]𝜑 ∧ 𝑥 = 𝑢)   )
43		sbequ1 2256	. . . . . . 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 44983	. . . . 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 44983	. . . 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 45044	. . 3 ⊢ (   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)   ▶   ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑)   )
50	49	in1 44927	. 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 1542  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-vd1 44926

This theorem is referenced by: (None)