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Theorem dedth2v 3945
Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3942 is simpler to use. See also comments in dedth 3941. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth2v.1  |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch )
)
dedth2v.2  |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch  <->  th )
)
dedth2v.3  |-  th
Assertion
Ref Expression
dedth2v  |-  ( ph  ->  ps )

Proof of Theorem dedth2v
StepHypRef Expression
1 dedth2v.1 . . 3  |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch )
)
2 dedth2v.2 . . 3  |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch  <->  th )
)
3 dedth2v.3 . . 3  |-  th
41, 2, 3dedth2h 3942 . 2  |-  ( (
ph  /\  ph )  ->  ps )
54anidms 645 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   ifcif 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-if 3892
This theorem is referenced by:  ltweuz  11887  omlsi  24944  pjhfo  25246  ghomgrplem  27444
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