Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm13.183 Structured version   Unicode version

Theorem pm13.183 3218
 Description: Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.183
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem pm13.183
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2433 . 2
2 eqeq2 2444 . . . 4
32bibi1d 320 . . 3
43albidv 1760 . 2
5 eqeq2 2444 . . . 4
65alrimiv 1766 . . 3
7 stdpc4 2148 . . . 4
8 sbbi 2196 . . . . 5
9 eqsb3 2549 . . . . . . 7
109bibi2i 314 . . . . . 6
11 equsb1 2161 . . . . . . 7
12 biimp 196 . . . . . . 7
1311, 12mpi 21 . . . . . 6
1410, 13sylbi 198 . . . . 5
158, 14sylbi 198 . . . 4
167, 15syl 17 . . 3
176, 16impbii 190 . 2
181, 4, 17vtoclbg 3146 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187  wal 1435   wceq 1437  wsb 1789   wcel 1870 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089 This theorem is referenced by:  mpt22eqb  6419
 Copyright terms: Public domain W3C validator