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Theorem jaob 857
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 555 . 2 (((𝜑𝜒) → 𝜓) → ((𝜑𝜓) ∧ (𝜒𝜓)))
5 pm3.44 534 . 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  863  pm5.53  872  pm4.83  1008  axio  2730  unss  3930  ralunb  3937  intun  4661  intpr  4662  relop  5428  sqrt2irr  15198  algcvgblem  15512  efgred  18381  caucfil  23301  plydivex  24271  2sqlem6  25368  arg-ax  32742  tendoeq2  36582  ifpidg  38356
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