Theorem jaoian 971

Description: Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)

Hypotheses

Ref	Expression
jaoian.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
jaoian.2	⊢ ((𝜃 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
jaoian	⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)

Proof of Theorem jaoian

Step	Hyp	Ref	Expression
1		jaoian.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 418	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		jaoian.2	. . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒)
4	3	ex 418	. . 3 ⊢ (𝜃 → (𝜓 → 𝜒))
5	2, 4	jaoi 871	. 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒))
6	5	imp 412	1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  ccase  1053  preq12nebg  4823  opthprneg  4825  elpreqpr  4827  tpres  7205  xaddnemnf  13359  xaddnepnf  13360  faclbnd  14427  faclbnd3  14429  faclbnd4lem1  14430  znf1o  21850  degltlem1  26383  ipasslem3  31428  padct  33303  fz1nntr  33387  xrge0iifhom  34562  bj-ideqg1ALT  38066  nn0addcom  43506  nn0mulcom  43510  fzsplit1nn0  43744  f1mo  49932