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Theorem cbvopab1 4496
 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 1754 . . . . 5
2 nfv 1754 . . . . . . 7
3 nfs1v 2233 . . . . . . 7
42, 3nfan 1986 . . . . . 6
54nfex 2006 . . . . 5
6 opeq1 4190 . . . . . . . 8
76eqeq2d 2443 . . . . . . 7
8 sbequ12 2049 . . . . . . 7
97, 8anbi12d 715 . . . . . 6
109exbidv 1761 . . . . 5
111, 5, 10cbvex 2078 . . . 4
12 nfv 1754 . . . . . . 7
13 cbvopab1.1 . . . . . . . 8
1413nfsb 2236 . . . . . . 7
1512, 14nfan 1986 . . . . . 6
1615nfex 2006 . . . . 5
17 nfv 1754 . . . . 5
18 opeq1 4190 . . . . . . . 8
1918eqeq2d 2443 . . . . . . 7
20 sbequ 2171 . . . . . . . 8
21 cbvopab1.2 . . . . . . . . 9
22 cbvopab1.3 . . . . . . . . 9
2321, 22sbie 2203 . . . . . . . 8
2420, 23syl6bb 264 . . . . . . 7
2519, 24anbi12d 715 . . . . . 6
2625exbidv 1761 . . . . 5
2716, 17, 26cbvex 2078 . . . 4
2811, 27bitri 252 . . 3
2928abbii 2563 . 2
30 df-opab 4485 . 2
31 df-opab 4485 . 2
3229, 30, 313eqtr4i 2468 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437  wex 1659  wnf 1663  wsb 1789  cab 2414  cop 4008  copab 4483 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-opab 4485 This theorem is referenced by:  cbvopab1v  4499  cbvmptf  4516  cbvmpt  4517  phpreu  31636  poimirlem26  31673  mbfposadd  31695
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