Theorem eloprabga 6389

Description: The law of concretion for operation class abstraction. Compare elopab 4760. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypothesis

Ref	Expression
eloprabga.1

Assertion

Ref	Expression
eloprabga

Distinct variable groups:   ,,,   ,,,   ,,,   ,,,

Proof of Theorem eloprabga

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		elex 3118	. 2
3		elex 3118	. 2
4		opex 4716	. . 3
5		simpr 461	. . . . . . . . . 10
6	5	eqeq1d 2459	. . . . . . . . 9
7		eqcom 2466	. . . . . . . . . 10
8		vex 3112	. . . . . . . . . . 11
9		vex 3112	. . . . . . . . . . 11
10		vex 3112	. . . . . . . . . . 11
11	8, 9, 10	otth2 4733	. . . . . . . . . 10
12	7, 11	bitri 249	. . . . . . . . 9
13	6, 12	syl6bb 261	. . . . . . . 8
14	13	anbi1d 704	. . . . . . 7
15		eloprabga.1	. . . . . . . 8
16	15	pm5.32i 637	. . . . . . 7
17	14, 16	syl6bb 261	. . . . . 6
18	17	3exbidv 1717	. . . . 5
19		df-oprab 6300	. . . . . . . . 9
20	19	eleq2i 2535	. . . . . . . 8
21		abid 2444	. . . . . . . 8
22	20, 21	bitr2i 250	. . . . . . 7
23		eleq1 2529	. . . . . . 7
24	22, 23	syl5bb 257	. . . . . 6
25	24	adantl 466	. . . . 5
26		elisset 3120	. . . . . . . . . 10
27		elisset 3120	. . . . . . . . . 10
28		elisset 3120	. . . . . . . . . 10
29	26, 27, 28	3anim123i 1181	. . . . . . . . 9
30		eeeanv 1989	. . . . . . . . 9
31	29, 30	sylibr 212	. . . . . . . 8
32	31	biantrurd 508	. . . . . . 7
33		19.41vvv 1773	. . . . . . 7
34	32, 33	syl6rbbr 264	. . . . . 6
35	34	adantr 465	. . . . 5
36	18, 25, 35	3bitr3d 283	. . . 4
37	36	expcom 435	. . 3
38	4, 37	vtocle 3183	. 2
39	1, 2, 3, 38	syl3an 1270	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  cvv 3109  <.cop 4035  {coprab 6297

This theorem is referenced by:  eloprabg  6390  ovigg  6423  vdwpc  14498  isrgra  24926  isrusgra  24927  elmpps  28933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-oprab 6300