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

Theorem caovcang 6248
 Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypothesis
Ref Expression
caovcang.1
Assertion
Ref Expression
caovcang
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem caovcang
StepHypRef Expression
1 caovcang.1 . . 3
21ralrimivvva 2799 . 2
3 oveq1 6088 . . . . 5
4 oveq1 6088 . . . . 5
53, 4eqeq12d 2450 . . . 4
65bibi1d 311 . . 3
7 oveq2 6089 . . . . 5
87eqeq1d 2444 . . . 4
9 eqeq1 2442 . . . 4
108, 9bibi12d 313 . . 3
11 oveq2 6089 . . . . 5
1211eqeq2d 2447 . . . 4
13 eqeq2 2445 . . . 4
1412, 13bibi12d 313 . . 3
156, 10, 14rspc3v 3061 . 2
162, 15mpan9 456 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  (class class class)co 6081 This theorem is referenced by:  caovcand  6249  caofcan  27517 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
 Copyright terms: Public domain W3C validator