Theorem jaoian 958

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 857	. 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒))
6	5	imp 406	1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847

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 207  df-an 396  df-or 848

This theorem is referenced by:  ccase  1037  preq12nebg  4815  opthprneg  4817  elpreqpr  4819  tpres  7135  xaddnemnf  13135  xaddnepnf  13136  faclbnd  14197  faclbnd3  14199  faclbnd4lem1  14200  znf1o  21489  degltlem1  26005  ipasslem3  30811  padct  32699  fz1nntr  32782  xrge0iifhom  33948  bj-ideqg1ALT  37205  nn0addcom  42501  nn0mulcom  42505  fzsplit1nn0  42793  f1mo  48890