Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  prodeq2w Unicode version

Theorem prodeq2w 11332
 Description: Equality theorem for product, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.)
Assertion
Ref Expression
prodeq2w
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem prodeq2w
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2139 . . . . . . . . . . . . 13
2 ifeq1 3477 . . . . . . . . . . . . . . 15
32alimi 1431 . . . . . . . . . . . . . 14
4 alral 2478 . . . . . . . . . . . . . 14
53, 4syl 14 . . . . . . . . . . . . 13
6 mpteq12 4011 . . . . . . . . . . . . 13
71, 5, 6sylancr 410 . . . . . . . . . . . 12
87seqeq3d 10233 . . . . . . . . . . 11
98breq1d 3939 . . . . . . . . . 10
109anbi2d 459 . . . . . . . . 9 # #
1110exbidv 1797 . . . . . . . 8 # #
1211rexbidv 2438 . . . . . . 7 # #
137seqeq3d 10233 . . . . . . . 8
1413breq1d 3939 . . . . . . 7
1512, 14anbi12d 464 . . . . . 6 # #
1615anbi2d 459 . . . . 5 DECID # DECID #
1716rexbidv 2438 . . . 4 DECID # DECID #
18 csbeq2 3026 . . . . . . . . . . . 12
1918ifeq1d 3489 . . . . . . . . . . 11
2019mpteq2dv 4019 . . . . . . . . . 10
2120seqeq3d 10233 . . . . . . . . 9
2221fveq1d 5423 . . . . . . . 8
2322eqeq2d 2151 . . . . . . 7
2423anbi2d 459 . . . . . 6
2524exbidv 1797 . . . . 5
2625rexbidv 2438 . . . 4
2717, 26orbi12d 782 . . 3 DECID # DECID #
2827iotabidv 5109 . 2 DECID # DECID #
29 df-proddc 11327 . 2 DECID #
30 df-proddc 11327 . 2 DECID #
3128, 29, 303eqtr4g 2197 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wo 697  DECID wdc 819  wal 1329   wceq 1331  wex 1468   wcel 1480  wral 2416  wrex 2417  csb 3003   wss 3071  cif 3474   class class class wbr 3929   cmpt 3989  cio 5086  wf1o 5122  cfv 5123  (class class class)co 5774  cc0 7627  c1 7628   cmul 7632   cle 7808   # cap 8350  cn 8727  cz 9061  cuz 9333  cfz 9797   cseq 10225   cli 11054  cprod 11326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10226  df-proddc 11327 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator