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Theorem cbvopab2 4271
 Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvopab2.1
cbvopab2.2
cbvopab2.3
Assertion
Ref Expression
cbvopab2
Distinct variable group:   ,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem cbvopab2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1629 . . . . . 6
2 cbvopab2.1 . . . . . 6
31, 2nfan 1846 . . . . 5
4 nfv 1629 . . . . . 6
5 cbvopab2.2 . . . . . 6
64, 5nfan 1846 . . . . 5
7 opeq2 3977 . . . . . . 7
87eqeq2d 2446 . . . . . 6
9 cbvopab2.3 . . . . . 6
108, 9anbi12d 692 . . . . 5
113, 6, 10cbvex 1983 . . . 4
1211exbii 1592 . . 3
1312abbii 2547 . 2
14 df-opab 4259 . 2
15 df-opab 4259 . 2
1613, 14, 153eqtr4i 2465 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550  wnf 1553   wceq 1652  cab 2421  cop 3809  copab 4257 This theorem is referenced by:  cbvoprab3  6140 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 2416 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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259
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