Theorem rimoveror-P4.31b 482

Description: '→' is Antidistributive on the Right Over '∨' . †

Assertion

Ref	Expression
rimoveror-P4.31b	⊢ (((𝜑 ∨ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜒) ∧ (𝜓 → 𝜒)))

Proof of Theorem rimoveror-P4.31b

Step	Hyp	Ref	Expression
1		sepimorl-P4.9b.CL 411	. 2 ⊢ (((𝜑 ∨ 𝜓) → 𝜒) → ((𝜑 → 𝜒) ∧ (𝜓 → 𝜒)))
2		joinimandinc-P4.8a.CL 399	. . 3 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → (𝜒 ∨ 𝜒)))
3		idempotor-P4.25b 451	. . . . 5 ⊢ ((𝜒 ∨ 𝜒) ↔ 𝜒)
4	3	subimr-P3.40b.RC 328	. . . 4 ⊢ (((𝜑 ∨ 𝜓) → (𝜒 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) → 𝜒))
5	4	rcp-NDBIEF0 240	. . 3 ⊢ (((𝜑 ∨ 𝜓) → (𝜒 ∨ 𝜒)) → ((𝜑 ∨ 𝜓) → 𝜒))
6	2, 5	syl-P3.24.RC 260	. 2 ⊢ (((𝜑 → 𝜒) ∧ (𝜓 → 𝜒)) → ((𝜑 ∨ 𝜓) → 𝜒))
7	1, 6	rcp-NDBII0 239	1 ⊢ (((𝜑 ∨ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜒) ∧ (𝜓 → 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

This theorem is referenced by: (None)