Theorem jaao 639

Description: Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)

Hypotheses

Ref	Expression
jaao.1	⊢ (𝜑 → (𝜓 → 𝜒))
jaao.2	⊢ (𝜃 → (𝜏 → 𝜒))

Assertion

Ref	Expression
jaao	⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒))

Proof of Theorem jaao

Step	Hyp	Ref	Expression
1		jaao.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	adantr 261	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒))
3		jaao.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜒))
4	3	adantl 262	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → 𝜒))
5	2, 4	jaod 637	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm3.48  699  prlem1  880  nford  1459  funun  4944  poxp  5853  nntri3or  6072