Theorem jaao 951

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 480	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒))
3		jaao.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜒))
4	3	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → 𝜒))
5	2, 4	jaod 855	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:  pm3.44  956  pm3.48  960  prlem1  1051  elpr2g  4582  ordtri1  6284  ordun  6352  suc11  6354  funun  6464  poxp  7940  suc11reg  9307  rankunb  9539  gruun  10493  ofpreima2  30905  wl-orel12  35597  clsk1indlem3  41542