Theorem orel 38110

Description: An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)

Hypotheses

Ref	Expression
orel.1	⊢ ((𝜓 ∧ 𝜂) → 𝜃)
orel.2	⊢ ((𝜒 ∧ 𝜌) → 𝜃)
orel.3	⊢ (𝜑 → (𝜓 ∨ 𝜒))

Assertion

Ref	Expression
orel	⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → 𝜃)

Proof of Theorem orel

Step	Hyp	Ref	Expression
1		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → 𝜂)
2		orel.1	. . . 4 ⊢ ((𝜓 ∧ 𝜂) → 𝜃)
3	2	ancoms 458	. . 3 ⊢ ((𝜂 ∧ 𝜓) → 𝜃)
4	1, 3	sylan 580	. 2 ⊢ (((𝜑 ∧ (𝜂 ∧ 𝜌)) ∧ 𝜓) → 𝜃)
5		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → 𝜌)
6		orel.2	. . . 4 ⊢ ((𝜒 ∧ 𝜌) → 𝜃)
7	6	ancoms 458	. . 3 ⊢ ((𝜌 ∧ 𝜒) → 𝜃)
8	5, 7	sylan 580	. 2 ⊢ (((𝜑 ∧ (𝜂 ∧ 𝜌)) ∧ 𝜒) → 𝜃)
9		orel.3	. . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
10	9	adantr 480	. 2 ⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → (𝜓 ∨ 𝜒))
11	4, 8, 10	mpjaodan 960	1 ⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847

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 207  df-an 396  df-or 848

This theorem is referenced by: (None)