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

Theorem opelopabaf 4776
 Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4774 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
opelopabaf.x
opelopabaf.y
opelopabaf.1
opelopabaf.2
opelopabaf.3
Assertion
Ref Expression
opelopabaf
Distinct variable groups:   ,,   ,,

Proof of Theorem opelopabaf
StepHypRef Expression
1 opelopabsb 4762 . 2
2 opelopabaf.1 . . 3
3 opelopabaf.2 . . 3
4 opelopabaf.x . . . 4
5 opelopabaf.y . . . 4
6 nfv 1707 . . . 4
7 opelopabaf.3 . . . 4
84, 5, 6, 7sbc2iegf 3402 . . 3
92, 3, 8mp2an 672 . 2
101, 9bitri 249 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/wnf 1616  e.wcel 1818   cvv 3109  [.wsbc 3327  <.cop 4035  {copab 4509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511
 Copyright terms: Public domain W3C validator