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Theorem cbvmpt2x 6375
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6376 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:   , , , ,   ,   ,

Proof of Theorem cbvmpt2x
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1707 . . . . 5
2 cbvmpt2x.1 . . . . . 6
32nfcri 2612 . . . . 5
41, 3nfan 1928 . . . 4
5 cbvmpt2x.3 . . . . 5
65nfeq2 2636 . . . 4
74, 6nfan 1928 . . 3
8 nfv 1707 . . . . 5
9 nfcv 2619 . . . . . 6
109nfcri 2612 . . . . 5
118, 10nfan 1928 . . . 4
12 cbvmpt2x.4 . . . . 5
1312nfeq2 2636 . . . 4
1411, 13nfan 1928 . . 3
15 nfv 1707 . . . . 5
16 cbvmpt2x.2 . . . . . 6
1716nfcri 2612 . . . . 5
1815, 17nfan 1928 . . . 4
19 cbvmpt2x.5 . . . . 5
2019nfeq2 2636 . . . 4
2118, 20nfan 1928 . . 3
22 nfv 1707 . . . 4
23 cbvmpt2x.6 . . . . 5
2423nfeq2 2636 . . . 4
2522, 24nfan 1928 . . 3
26 eleq1 2529 . . . . . 6
2726adantr 465 . . . . 5
28 cbvmpt2x.7 . . . . . . 7
2928eleq2d 2527 . . . . . 6
30 eleq1 2529 . . . . . 6
3129, 30sylan9bb 699 . . . . 5
3227, 31anbi12d 710 . . . 4
33 cbvmpt2x.8 . . . . 5
3433eqeq2d 2471 . . . 4
3532, 34anbi12d 710 . . 3
367, 14, 21, 25, 35cbvoprab12 6371 . 2
37 df-mpt2 6301 . 2
38 df-mpt2 6301 . 2
3936, 37, 383eqtr4i 2496 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  F/_wnfc 2605  {coprab 6297  e.cmpt2 6298
This theorem is referenced by:  cbvmpt2  6376  mpt2mptsx  6863  dmmpt2ssx  6865  gsumcom2  17003  ptcmpg  20557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-oprab 6300  df-mpt2 6301
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