Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  keephyp2v Structured version   Unicode version

Theorem keephyp2v 3796
 Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3782). (Contributed by NM, 16-Apr-2005.)
Hypotheses
Ref Expression
keephyp2v.1
keephyp2v.2
keephyp2v.3
keephyp2v.4
keephyp2v.5
keephyp2v.6
Assertion
Ref Expression
keephyp2v

Proof of Theorem keephyp2v
StepHypRef Expression
1 keephyp2v.5 . . 3
2 iftrue 3747 . . . . . 6
32eqcomd 2443 . . . . 5
4 keephyp2v.1 . . . . 5
53, 4syl 16 . . . 4
6 iftrue 3747 . . . . . 6
76eqcomd 2443 . . . . 5
8 keephyp2v.2 . . . . 5
97, 8syl 16 . . . 4
105, 9bitrd 246 . . 3
111, 10mpbii 204 . 2
12 keephyp2v.6 . . 3
13 iffalse 3748 . . . . . 6
1413eqcomd 2443 . . . . 5
15 keephyp2v.3 . . . . 5
1614, 15syl 16 . . . 4
17 iffalse 3748 . . . . . 6
1817eqcomd 2443 . . . . 5
19 keephyp2v.4 . . . . 5
2018, 19syl 16 . . . 4
2116, 20bitrd 246 . . 3
2212, 21mpbii 204 . 2
2311, 22pm2.61i 159 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wceq 1653  cif 3741 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-if 3742
 Copyright terms: Public domain W3C validator