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Theorem dedth2v 3786
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 3783 is simpler to use. See also comments in dedth 3782. (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 3783 . 2  |-  ( (
ph  /\  ph )  ->  ps )
54anidms 628 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653   ifcif 3741
This theorem is referenced by:  ltweuz  11303  omlsi  22908  pjhfo  23210  ghomgrplem  25102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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