Theorem jaoian 807

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 115	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		jaoian.2	. . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒)
4	3	ex 115	. . 3 ⊢ (𝜃 → (𝜓 → 𝜒))
5	2, 4	jaoi 728	. 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒))
6	5	imp 124	1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  ordi  828  ccase  977  dfifp2dc  994  xaddnemnf  10270  xaddnepnf  10271  flaplt  10733  faclbnd  11195  faclbnd3  11197  znf1o  15070