Theorem jaao 953

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 858	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846

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 847

This theorem is referenced by:  pm3.44  958  pm3.48  962  prlem1  1053  elpr2g  4648  ordtri1  6396  ordun  6467  suc11  6470  funun  6593  poxp  8127  suc11reg  9636  rankunb  9867  gruun  10823  ofpreima2  32445  wl-orel12  36972  clsk1indlem3  43467