Theorem jaoa 952

Description: Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)

Hypotheses

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

Assertion

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

Proof of Theorem jaoa

Step	Hyp	Ref	Expression
1		jaao.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	adantrd 491	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜒))
3		jaao.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜒))
4	3	adantld 490	. 2 ⊢ (𝜃 → ((𝜓 ∧ 𝜏) → 𝜒))
5	2, 4	jaoi 853	1 ⊢ ((𝜑 ∨ 𝜃) → ((𝜓 ∧ 𝜏) → 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843

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 206  df-an 396  df-or 844

This theorem is referenced by:  pm4.79  1000  19.40b  1892  2eu3  2655  abslt  14954  absle  14955  unconn  22488  dfon2lem4  33668  clsk1indlem3  41542