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Theorem cbvopab1 3858
 Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbvopab1.1
cbvopab1.2
cbvopab1.3
Assertion
Ref Expression
cbvopab1
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem cbvopab1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . . 5
2 nfv 1437 . . . . . . 7
3 nfs1v 1831 . . . . . . 7
42, 3nfan 1473 . . . . . 6
54nfex 1544 . . . . 5
6 opeq1 3577 . . . . . . . 8
76eqeq2d 2067 . . . . . . 7
8 sbequ12 1670 . . . . . . 7
97, 8anbi12d 450 . . . . . 6
109exbidv 1722 . . . . 5
111, 5, 10cbvex 1655 . . . 4
12 nfv 1437 . . . . . . 7
13 cbvopab1.1 . . . . . . . 8
1413nfsb 1838 . . . . . . 7
1512, 14nfan 1473 . . . . . 6
1615nfex 1544 . . . . 5
17 nfv 1437 . . . . 5
18 opeq1 3577 . . . . . . . 8
1918eqeq2d 2067 . . . . . . 7
20 sbequ 1737 . . . . . . . 8
21 cbvopab1.2 . . . . . . . . 9
22 cbvopab1.3 . . . . . . . . 9
2321, 22sbie 1690 . . . . . . . 8
2420, 23syl6bb 189 . . . . . . 7
2519, 24anbi12d 450 . . . . . 6
2625exbidv 1722 . . . . 5
2716, 17, 26cbvex 1655 . . . 4
2811, 27bitri 177 . . 3
2928abbii 2169 . 2
30 df-opab 3847 . 2
31 df-opab 3847 . 2
3229, 30, 313eqtr4i 2086 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wb 102   wceq 1259  wnf 1365  wex 1397  wsb 1661  cab 2042  cop 3406  copab 3845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847 This theorem is referenced by:  cbvopab1v  3861  cbvmpt  3879
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