Theorem jaoian 953

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 412	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		jaoian.2	. . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒)
4	3	ex 412	. . 3 ⊢ (𝜃 → (𝜓 → 𝜒))
5	2, 4	jaoi 854	. 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒))
6	5	imp 406	1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844

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 206  df-an 396  df-or 845

This theorem is referenced by:  ccase  1034  preq12nebg  4856  opthprneg  4858  elpreqpr  4860  tpres  7195  xaddnemnf  13213  xaddnepnf  13214  faclbnd  14248  faclbnd3  14250  faclbnd4lem1  14251  znf1o  21416  degltlem1  25932  ipasslem3  30558  padct  32416  fz1nntr  32487  xrge0iifhom  33409  bj-ideqg1ALT  36537  nn0addcom  41837  nn0mulcom  41841  fzsplit1nn0  42006  f1mo  47731