Theorem cbvoprab12 6371

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Hypotheses

Ref	Expression
cbvoprab12.1
cbvoprab12.2
cbvoprab12.3
cbvoprab12.4
cbvoprab12.5

Assertion

Ref	Expression
cbvoprab12

Distinct variable group:   ,,,,

Proof of Theorem cbvoprab12

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1707	. . . . 5
2		cbvoprab12.1	. . . . 5
3	1, 2	nfan 1928	. . . 4
4		nfv 1707	. . . . 5
5		cbvoprab12.2	. . . . 5
6	4, 5	nfan 1928	. . . 4
7		nfv 1707	. . . . 5
8		cbvoprab12.3	. . . . 5
9	7, 8	nfan 1928	. . . 4
10		nfv 1707	. . . . 5
11		cbvoprab12.4	. . . . 5
12	10, 11	nfan 1928	. . . 4
13		opeq12 4219	. . . . . 6
14	13	eqeq2d 2471	. . . . 5
15		cbvoprab12.5	. . . . 5
16	14, 15	anbi12d 710	. . . 4
17	3, 6, 9, 12, 16	cbvex2 2028	. . 3
18	17	opabbii 4516	. 2
19		dfoprab2 6343	. 2
20		dfoprab2 6343	. 2
21	18, 19, 20	3eqtr4i 2496	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  F/wnf 1616  <.cop 4035  {copab 4509  {coprab 6297

This theorem is referenced by:  cbvoprab12v  6372  cbvmpt2x  6375  dfoprab4f  6858  fmpt2x  6866  tposoprab  7010  cbvmpt2x2  32925

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-opab 4511  df-oprab 6300