Theorem jaoian 819

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 448	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		jaoian.2	. . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒)
4	3	ex 448	. . 3 ⊢ (𝜃 → (𝜓 → 𝜒))
5	2, 4	jaoi 392	. 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒))
6	5	imp 443	1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∨ wo 381   ∧ wa 382

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 195  df-or 383  df-an 384

This theorem is referenced by:  ccase  983  elpreqpr  4329  tpres  6348  xaddnemnf  11901  xaddnepnf  11902  faclbnd  12896  faclbnd3  12898  faclbnd4lem1  12899  znf1o  19666  degltlem1  23580  ipasslem3  26865  padct  28678  fz1nntr  28741  xrge0iifhom  29104  fzsplit1nn0  36118