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Theorem cbvopab1 4281
 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 1630 . . . . 5
2 nfv 1630 . . . . . . 7
3 nfs1v 2184 . . . . . . 7
42, 3nfan 1847 . . . . . 6
54nfex 1866 . . . . 5
6 opeq1 3986 . . . . . . . 8
76eqeq2d 2449 . . . . . . 7
8 sbequ12 1945 . . . . . . 7
97, 8anbi12d 693 . . . . . 6
109exbidv 1637 . . . . 5
111, 5, 10cbvex 1984 . . . 4
12 nfv 1630 . . . . . . 7
13 cbvopab1.1 . . . . . . . 8
1413nfsb 2187 . . . . . . 7
1512, 14nfan 1847 . . . . . 6
1615nfex 1866 . . . . 5
17 nfv 1630 . . . . 5
18 opeq1 3986 . . . . . . . 8
1918eqeq2d 2449 . . . . . . 7
20 sbequ 2113 . . . . . . . 8
21 cbvopab1.2 . . . . . . . . 9
22 cbvopab1.3 . . . . . . . . 9
2321, 22sbie 2152 . . . . . . . 8
2420, 23syl6bb 254 . . . . . . 7
2519, 24anbi12d 693 . . . . . 6
2625exbidv 1637 . . . . 5
2716, 17, 26cbvex 1984 . . . 4
2811, 27bitri 242 . . 3
2928abbii 2550 . 2
30 df-opab 4270 . 2
31 df-opab 4270 . 2
3229, 30, 313eqtr4i 2468 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551  wnf 1554   wceq 1653  wsb 1659  cab 2424  cop 3819  copab 4268 This theorem is referenced by:  cbvopab1v  4284  cbvmpt  4302  cbvmptf  24073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270
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