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Theorem cbvopab 4521
 Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1
cbvopab.2
cbvopab.3
cbvopab.4
cbvopab.5
Assertion
Ref Expression
cbvopab
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)   (,,,)

Proof of Theorem cbvopab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1683 . . . . 5
2 cbvopab.1 . . . . 5
31, 2nfan 1875 . . . 4
4 nfv 1683 . . . . 5
5 cbvopab.2 . . . . 5
64, 5nfan 1875 . . . 4
7 nfv 1683 . . . . 5
8 cbvopab.3 . . . . 5
97, 8nfan 1875 . . . 4
10 nfv 1683 . . . . 5
11 cbvopab.4 . . . . 5
1210, 11nfan 1875 . . . 4
13 opeq12 4221 . . . . . 6
1413eqeq2d 2481 . . . . 5
15 cbvopab.5 . . . . 5
1614, 15anbi12d 710 . . . 4
173, 6, 9, 12, 16cbvex2 2001 . . 3
1817abbii 2601 . 2
19 df-opab 4512 . 2
20 df-opab 4512 . 2
2118, 19, 203eqtr4i 2506 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379  wex 1596  wnf 1599  cab 2452  cop 4039  copab 4510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512 This theorem is referenced by:  cbvopabv  4522  dfrel4  27277  aomclem8  30935
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