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Theorem dedth 3703
 Description: Weak deduction theorem that eliminates a hypothesis , making it become an antecedent. We assume that a proof exists for when the class variable is replaced with a specific class . The hypothesis should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3710. If the inference has other hypotheses with class variable , these can be kept by assigning keephyp 3716 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpeuni/mmdeduction.html 3716. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
dedth.1
dedth.2
Assertion
Ref Expression
dedth

Proof of Theorem dedth
StepHypRef Expression
1 dedth.2 . 2
2 iftrue 3668 . . . 4
32eqcomd 2358 . . 3
4 dedth.1 . . 3
53, 4syl 15 . 2
61, 5mpbiri 224 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wceq 1642  cif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663 This theorem is referenced by:  dedth2h  3704  dedth3h  3705
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