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Theorem cbvprodi 25312
 Description: Change bound variable in a finite composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
Hypotheses
Ref Expression
cbvprod.1
cbvprod.2
cbvprod.3
Assertion
Ref Expression
cbvprodi

Proof of Theorem cbvprodi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvprod.1 . . . . . . . . . . 11
2 cbvprod.2 . . . . . . . . . . 11
3 cbvprod.3 . . . . . . . . . . 11
41, 2, 3cbvmpt 4110 . . . . . . . . . 10
5 seqeq3 11051 . . . . . . . . . 10
64, 5ax-mp 8 . . . . . . . . 9
76fveq1i 5526 . . . . . . . 8
87eleq2i 2347 . . . . . . 7
98anbi2i 675 . . . . . 6
109rexbii 2568 . . . . 5
1110exbii 1569 . . . 4
1211abbii 2395 . . 3
13 ifeq2 3570 . . 3 GId GId
1412, 13ax-mp 8 . 2 GId GId
15 df-prod 25299 . 2 GId
16 df-prod 25299 . 2 GId
1714, 15, 163eqtr4i 2313 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1528   wceq 1623   wcel 1684  cab 2269  wnfc 2406  wrex 2544  cvv 2788  c0 3455  cif 3565   cmpt 4077  cfv 5255  (class class class)co 5858  cuz 10230  cfz 10782   cseq 11046  GIdcgi 20854  cprd 25298 This theorem is referenced by:  prodeqfv  25318  fprodserf  25321  fprod1s  25325  fprodp1s  25327 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-prod 25299
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