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Theorem prodeq1 25409
 Description: Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
prodeq1

Proof of Theorem prodeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2302 . . 3
2 eqeq1 2302 . . . . . . 7
32anbi1d 685 . . . . . 6
43rexbidv 2577 . . . . 5
54exbidv 1616 . . . 4
65abbidv 2410 . . 3
71, 6ifbieq2d 3598 . 2 GId GId
8 df-prod 25402 . 2 GId
9 df-prod 25402 . 2 GId
107, 8, 93eqtr4g 2353 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1531   wceq 1632   wcel 1696  cab 2282  wrex 2557  cvv 2801  c0 3468  cif 3578   cmpt 4093  cfv 5271  (class class class)co 5874  cuz 10246  cfz 10798   cseq 11062  GIdcgi 20870  cprd 25401 This theorem is referenced by:  prodeq1d  25416  fsumprd  25432 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-prod 25402
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