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Theorem cbvoprab1 5567
 Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1
cbvoprab1.2
cbvoprab1.3
Assertion
Ref Expression
cbvoprab1
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)   (,,,)

Proof of Theorem cbvoprab1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . . 6
2 cbvoprab1.1 . . . . . 6
31, 2nfan 1824 . . . . 5
43nfex 1843 . . . 4
5 nfv 1619 . . . . . 6
6 cbvoprab1.2 . . . . . 6
75, 6nfan 1824 . . . . 5
87nfex 1843 . . . 4
9 opeq1 4578 . . . . . . 7
109eqeq2d 2364 . . . . . 6
11 cbvoprab1.3 . . . . . 6
1210, 11anbi12d 691 . . . . 5
1312exbidv 1626 . . . 4
144, 8, 13cbvex 1985 . . 3
1514opabbii 4626 . 2
16 dfoprab2 5558 . 2
17 dfoprab2 5558 . 2
1815, 16, 173eqtr4i 2383 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wex 1541  wnf 1544   wceq 1642  cop 4561  copab 4622  coprab 5527 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-oprab 5528 This theorem is referenced by: (None)
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