Theorem oroverbiint-P4.28d 471

Description: One Direction of '∨' Distributes Over '↔'. †

Only this version is deducible with intuitionist logic.

Assertion

Ref	Expression
oroverbiint-P4.28d	⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒)))

Proof of Theorem oroverbiint-P4.28d

Step	Hyp	Ref	Expression
1		oroverbi-P4.28-L3 468	. . . . 5 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ (𝜓 → 𝜒)) ∧ (𝜑 ∨ (𝜒 → 𝜓))))
2	1	rcp-NDBIEF0 240	. . . 4 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) → ((𝜑 ∨ (𝜓 → 𝜒)) ∧ (𝜑 ∨ (𝜒 → 𝜓))))
3	2	ndander-P3.9 174	. . 3 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒)))
4		oroverim-P4.28-L1 465	. . 3 ⊢ ((𝜑 ∨ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
5	3, 4	syl-P3.24.RC 260	. 2 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
6	2	ndandel-P3.8 173	. . 3 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) → (𝜑 ∨ (𝜒 → 𝜓)))
7		oroverim-P4.28-L1 465	. . 3 ⊢ ((𝜑 ∨ (𝜒 → 𝜓)) → ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜓)))
8	6, 7	syl-P3.24.RC 260	. 2 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) → ((𝜑 ∨ 𝜒) → (𝜑 ∨ 𝜓)))
9	5, 8	ndbii-P3.13 178	1 ⊢ ((𝜑 ∨ (𝜓 ↔ 𝜒)) → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∧ 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  df-rcp-AND3 161  df-rcp-AND4 163

This theorem is referenced by: (None)