Theorem jaao 969

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 486	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒))
3		jaao.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜒))
4	3	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → 𝜒))
5	2, 4	jaod 873	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  pm3.44  974  pm3.48  978  prlem1  1070  elpr2g  4606  ordtri1  6399  ordun  6472  suc11  6475  funun  6591  poxp  8146  suc11reg  9627  rankunb  9871  gruun  10898  ofpreima2  33311  wl-orel12  38552  clsk1indlem3  45152