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Theorem pm5.21ndd 621
 Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)
Hypotheses
Ref Expression
pm5.21ndd.1 (𝜑 → (𝜒𝜓))
pm5.21ndd.2 (𝜑 → (𝜃𝜓))
pm5.21ndd.3 (𝜑 → (𝜓 → (𝜒𝜃)))
Assertion
Ref Expression
pm5.21ndd (𝜑 → (𝜒𝜃))

Proof of Theorem pm5.21ndd
StepHypRef Expression
1 pm5.21ndd.1 . . . 4 (𝜑 → (𝜒𝜓))
2 pm5.21ndd.3 . . . 4 (𝜑 → (𝜓 → (𝜒𝜃)))
31, 2syld 40 . . 3 (𝜑 → (𝜒 → (𝜒𝜃)))
43ibd 167 . 2 (𝜑 → (𝜒𝜃))
5 pm5.21ndd.2 . . . . 5 (𝜑 → (𝜃𝜓))
65, 2syld 40 . . . 4 (𝜑 → (𝜃 → (𝜒𝜃)))
7 bicom1 122 . . . 4 ((𝜒𝜃) → (𝜃𝜒))
86, 7syl6 29 . . 3 (𝜑 → (𝜃 → (𝜃𝜒)))
98ibd 167 . 2 (𝜑 → (𝜃𝜒))
104, 9impbid 120 1 (𝜑 → (𝜒𝜃))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  pm5.21nd  825  sbcrext  2835  rmob  2850  epelg  4027  eqbrrdva  4505  relbrcnvg  4704  fmptco  5330  ovelrn  5649  brtpos2  5866  brdomg  6229  genpelvl  6610  genpelvu  6611  fzoval  9005  clim  9802
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