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Theorem cbvopab 4276
 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 1629 . . . . 5
2 cbvopab.1 . . . . 5
31, 2nfan 1846 . . . 4
4 nfv 1629 . . . . 5
5 cbvopab.2 . . . . 5
64, 5nfan 1846 . . . 4
7 nfv 1629 . . . . 5
8 cbvopab.3 . . . . 5
97, 8nfan 1846 . . . 4
10 nfv 1629 . . . . 5
11 cbvopab.4 . . . . 5
1210, 11nfan 1846 . . . 4
13 opeq12 3986 . . . . . 6
1413eqeq2d 2447 . . . . 5
15 cbvopab.5 . . . . 5
1614, 15anbi12d 692 . . . 4
173, 6, 9, 12, 16cbvex2 1991 . . 3
1817abbii 2548 . 2
19 df-opab 4267 . 2
20 df-opab 4267 . 2
2118, 19, 203eqtr4i 2466 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550  wnf 1553   wceq 1652  cab 2422  cop 3817  copab 4265 This theorem is referenced by:  cbvopabv  4277  dfrel4  24034  aomclem8  27136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267
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