Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvmpt2x Structured version   Unicode version

Theorem cbvmpt2x 6142
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6143 allows to be a function of . (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpt2x.1
cbvmpt2x.2
cbvmpt2x.3
cbvmpt2x.4
cbvmpt2x.5
cbvmpt2x.6
cbvmpt2x.7
cbvmpt2x.8
Assertion
Ref Expression
cbvmpt2x
Distinct variable groups:   ,,,,   ,   ,
Allowed substitution hints:   (,,)   (,,,)   (,,)   (,,,)

Proof of Theorem cbvmpt2x
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1629 . . . . 5
2 cbvmpt2x.1 . . . . . 6
32nfcri 2565 . . . . 5
41, 3nfan 1846 . . . 4
5 cbvmpt2x.3 . . . . 5
65nfeq2 2582 . . . 4
74, 6nfan 1846 . . 3
8 nfv 1629 . . . . 5
9 nfcv 2571 . . . . . 6
109nfcri 2565 . . . . 5
118, 10nfan 1846 . . . 4
12 cbvmpt2x.4 . . . . 5
1312nfeq2 2582 . . . 4
1411, 13nfan 1846 . . 3
15 nfv 1629 . . . . 5
16 cbvmpt2x.2 . . . . . 6
1716nfcri 2565 . . . . 5
1815, 17nfan 1846 . . . 4
19 cbvmpt2x.5 . . . . 5
2019nfeq2 2582 . . . 4
2118, 20nfan 1846 . . 3
22 nfv 1629 . . . 4
23 cbvmpt2x.6 . . . . 5
2423nfeq2 2582 . . . 4
2522, 24nfan 1846 . . 3
26 eleq1 2495 . . . . . 6
2726adantr 452 . . . . 5
28 cbvmpt2x.7 . . . . . . 7
2928eleq2d 2502 . . . . . 6
30 eleq1 2495 . . . . . 6
3129, 30sylan9bb 681 . . . . 5
3227, 31anbi12d 692 . . . 4
33 cbvmpt2x.8 . . . . 5
3433eqeq2d 2446 . . . 4
3532, 34anbi12d 692 . . 3
367, 14, 21, 25, 35cbvoprab12 6138 . 2
37 df-mpt2 6078 . 2
38 df-mpt2 6078 . 2
3936, 37, 383eqtr4i 2465 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wnfc 2558  coprab 6074   cmpt2 6075 This theorem is referenced by:  cbvmpt2  6143  mpt2mptsx  6406  dmmpt2ssx  6408  gsumcom2  15541  ptcmpg  18080 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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-ne 2600  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  df-oprab 6077  df-mpt2 6078
 Copyright terms: Public domain W3C validator