Theorem joinimandinc-P4.8a 397

Description: Join Two Implications Through Conjunction (Inclusive). †

Hypothesis

Ref	Expression
joinimandinc-P4.8a.1	⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜗)))

Assertion

Ref	Expression
joinimandinc-P4.8a	⊢ (𝛾 → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜗)))

Proof of Theorem joinimandinc-P4.8a

Step	Hyp	Ref	Expression
1		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → 𝜑)
2		joinimandinc-P4.8a.1	. . . . . . 7 ⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜗)))
3	2	rcp-NDIMP1add2 212	. . . . . 6 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜗)))
4	3	ndander-P3.9 174	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → (𝜑 → 𝜓))
5	1, 4	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → 𝜓)
6	5	ndorir-P3.11 176	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜑) → (𝜓 ∨ 𝜗))
7		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜒) → 𝜒)
8	2	rcp-NDIMP1add2 212	. . . . . 6 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜒) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜗)))
9	8	ndandel-P3.8 173	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜒) → (𝜒 → 𝜗))
10	7, 9	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜒) → 𝜗)
11	10	ndoril-P3.10 175	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒) ∧ 𝜒) → (𝜓 ∨ 𝜗))
12		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒)) → (𝜑 ∨ 𝜒))
13	6, 11, 12	rcp-NDORE3 236	. 2 ⊢ ((𝛾 ∧ (𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜗))
14	13	rcp-NDIMI2 224	1 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜗)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144   ∧ wff-rcp-AND3 160

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-rcp-AND3 161

This theorem is referenced by:  joinimandinc-P4.8a.RC  398  joinimandinc-P4.8a.CL  399  oroverim-P4.28-L1  465  joinimandinc2-P4  576  joinimandinc3-P4  578