Theorem joinimor-P4.8c 403

Description: Join Two Implications Through Disjunction. †

Hypothesis

Ref	Expression
joinimor-P4.8c.1	⊢ (𝛾 → ((𝜑 → 𝜓) ∨ (𝜒 → 𝜗)))

Assertion

Ref	Expression
joinimor-P4.8c	⊢ (𝛾 → ((𝜑 ∧ 𝜒) → (𝜓 ∨ 𝜗)))

Proof of Theorem joinimor-P4.8c

Step	Hyp	Ref	Expression
1		rcp-NDASM2of3 196	. . . . . 6 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 → 𝜓)) → (𝜑 ∧ 𝜒))
2	1	ndander-P3.9 174	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 → 𝜓)) → 𝜑)
3		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 → 𝜓)) → (𝜑 → 𝜓))
4	2, 3	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 → 𝜓)) → 𝜓)
5	4	ndorir-P3.11 176	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 → 𝜓)) → (𝜓 ∨ 𝜗))
6		rcp-NDASM2of3 196	. . . . . 6 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜒 → 𝜗)) → (𝜑 ∧ 𝜒))
7	6	ndandel-P3.8 173	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜒 → 𝜗)) → 𝜒)
8		rcp-NDASM3of3 197	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜒 → 𝜗)) → (𝜒 → 𝜗))
9	7, 8	ndime-P3.6 171	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜒 → 𝜗)) → 𝜗)
10	9	ndoril-P3.10 175	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒) ∧ (𝜒 → 𝜗)) → (𝜓 ∨ 𝜗))
11		joinimor-P4.8c.1	. . . 4 ⊢ (𝛾 → ((𝜑 → 𝜓) ∨ (𝜒 → 𝜗)))
12	11	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → ((𝜑 → 𝜓) ∨ (𝜒 → 𝜗)))
13	5, 10, 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:  joinimor-P4.8c.RC  404  joinimor-P4.8c.CL  405  oroverim-P4.28-L1  465  joinimor2-P4  584  joinimor3-P4  586