Theorem jaoian 824

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 450	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		jaoian.2	. . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒)
4	3	ex 450	. . 3 ⊢ (𝜃 → (𝜓 → 𝜒))
5	2, 4	jaoi 394	. 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒))
6	5	imp 445	1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  ccase  987  elpreqpr  4396  tpres  6466  xaddnemnf  12067  xaddnepnf  12068  faclbnd  13077  faclbnd3  13079  faclbnd4lem1  13080  znf1o  19900  degltlem1  23832  ipasslem3  27688  padct  29497  fz1nntr  29561  xrge0iifhom  29983  fzsplit1nn0  37317