Theorem 3jaoi 1282

Description: Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)

Hypotheses

Ref	Expression
3jaoi.1	⊢ (𝜑 → 𝜓)
3jaoi.2	⊢ (𝜒 → 𝜓)
3jaoi.3	⊢ (𝜃 → 𝜓)

Assertion

Ref	Expression
3jaoi	⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)

Proof of Theorem 3jaoi

Step	Hyp	Ref	Expression
1		3jaoi.1	. . 3 ⊢ (𝜑 → 𝜓)
2		3jaoi.2	. . 3 ⊢ (𝜒 → 𝜓)
3		3jaoi.3	. . 3 ⊢ (𝜃 → 𝜓)
4	1, 2, 3	3pm3.2i 1160	. 2 ⊢ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))
5		3jao 1280	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) → ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓))
6	4, 5	ax-mp 5	1 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)

Syntax hints:   → wi 4   ∨ w3o 962   ∧ w3a 963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965

This theorem is referenced by:  3jaoian  1284  3ianorr  1288  acexmidlem1  5810  nndceq  6435  nndcel  6436  znegcl  9177  xrltnr  9664  nltpnft  9696  ngtmnft  9699  xrrebnd  9701  xnegcl  9714  xnegneg  9715  xltnegi  9717  xrpnfdc  9724  xrmnfdc  9725  xnegid  9741  xaddid1  9744  xposdif  9764