Theorem andoveror-P4.27-L2 460

Description: Lemma for andoveror-P4.27a 461. †

Assertion

Ref	Expression
andoveror-P4.27-L2	⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒)))

Proof of Theorem andoveror-P4.27-L2

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜓)) → (𝜑 ∧ 𝜓))
2	1	ndander-P3.9 174	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜓)) → 𝜑)
3	1	ndandel-P3.8 173	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜓)) → 𝜓)
4	3	ndorir-P3.11 176	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜓)) → (𝜓 ∨ 𝜒))
5	2, 4	ndandi-P3.7 172	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜓)) → (𝜑 ∧ (𝜓 ∨ 𝜒)))
6		rcp-NDASM2of2 194	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜒)) → (𝜑 ∧ 𝜒))
7	6	ndander-P3.9 174	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜒)) → 𝜑)
8	6	ndandel-P3.8 173	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜒)) → 𝜒)
9	8	ndoril-P3.10 175	. . 3 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜒)) → (𝜓 ∨ 𝜒))
10	7, 9	ndandi-P3.7 172	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) ∧ (𝜑 ∧ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒)))
11		rcp-NDASM1of1 192	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
12	5, 10, 11	rcp-NDORE2 235	1 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155

This theorem is referenced by:  andoveror-P4.27a  461