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Theorem cbvopab 4520
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:   , , ,

Proof of Theorem cbvopab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1707 . . . . 5
2 cbvopab.1 . . . . 5
31, 2nfan 1928 . . . 4
4 nfv 1707 . . . . 5
5 cbvopab.2 . . . . 5
64, 5nfan 1928 . . . 4
7 nfv 1707 . . . . 5
8 cbvopab.3 . . . . 5
97, 8nfan 1928 . . . 4
10 nfv 1707 . . . . 5
11 cbvopab.4 . . . . 5
1210, 11nfan 1928 . . . 4
13 opeq12 4219 . . . . . 6
1413eqeq2d 2471 . . . . 5
15 cbvopab.5 . . . . 5
1614, 15anbi12d 710 . . . 4
173, 6, 9, 12, 16cbvex2 2028 . . 3
1817abbii 2591 . 2
19 df-opab 4511 . 2
20 df-opab 4511 . 2
2118, 19, 203eqtr4i 2496 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  F/wnf 1616  {cab 2442  <.cop 4035  {copab 4509
This theorem is referenced by:  cbvopabv  4521  dfrel4  27454  aomclem8  31007
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
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-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
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