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Theorem 3jaoi 1198
 Description: Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
Hypotheses
Ref Expression
3jaoi.1 (𝜑𝜓)
3jaoi.2 (𝜒𝜓)
3jaoi.3 (𝜃𝜓)
Assertion
Ref Expression
3jaoi ((𝜑𝜒𝜃) → 𝜓)

Proof of Theorem 3jaoi
StepHypRef Expression
1 3jaoi.1 . . 3 (𝜑𝜓)
2 3jaoi.2 . . 3 (𝜒𝜓)
3 3jaoi.3 . . 3 (𝜃𝜓)
41, 2, 33pm3.2i 1082 . 2 ((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓))
5 3jao 1196 . 2 (((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))
64, 5ax-mp 7 1 ((𝜑𝜒𝜃) → 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ w3o 884   ∧ w3a 885 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630 This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887 This theorem is referenced by:  3jaoian  1200  3ianorr  1204  sspsstrir  3046  acexmidlem1  5508  nndceq  6077  nndcel  6078  znegcl  8276  xrltnr  8701  nltpnft  8730  ngtmnft  8731  xrrebnd  8732  xnegcl  8745  xnegneg  8746  xltnegi  8748
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