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  4817  opthprneg  4819  elpreqpr  4821  tpres  7141  xaddnemnf  13156  xaddnepnf  13157  faclbnd  14215  faclbnd3  14217  faclbnd4lem1  14218  znf1o  21476  degltlem1  25993  ipasslem3  30795  padct  32676  fz1nntr  32760  xrge0iifhom  33906  bj-ideqg1ALT  37141  nn0addcom  42438  nn0mulcom  42442  fzsplit1nn0  42730  f1mo  48841