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Theorem jaob 839
Description: Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
Assertion
Ref Expression
jaob (((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Proof of Theorem jaob
StepHypRef Expression
1 pm2.67-2 416 . . 3 (((𝜑𝜒) → 𝜓) → (𝜑𝜓))
2 olc 398 . . . 4 (𝜒 → (𝜑𝜒))
32imim1i 63 . . 3 (((𝜑𝜒) → 𝜓) → (𝜒𝜓))
41, 3jca 553 . 2 (((𝜑𝜒) → 𝜓) → ((𝜑𝜓) ∧ (𝜒𝜓)))
5 pm3.44 532 . 2 (((𝜑𝜓) ∧ (𝜒𝜓)) → ((𝜑𝜒) → 𝜓))
64, 5impbii 199 1 (((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385
This theorem is referenced by:  pm4.77  845  pm5.53  854  pm4.83  990  axio  2621  unss  3820  ralunb  3827  intun  4541  intpr  4542  relop  5305  sqrt2irr  15023  algcvgblem  15337  efgred  18207  caucfil  23127  plydivex  24097  2sqlem6  25193  arg-ax  32540  tendoeq2  36379  ifpidg  38153
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