Theorem rimoverand-P4.31-L1 480

Description: Lemma for rimoverand-P4.31a 481 and rimoverandint-P4.31c 483. †

Assertion

Ref	Expression
rimoverand-P4.31-L1	⊢ (((𝜑 → 𝜒) ∨ (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → 𝜒))

Proof of Theorem rimoverand-P4.31-L1

Step	Hyp	Ref	Expression
1		joinimor-P4.8c.CL 405	. 2 ⊢ (((𝜑 → 𝜒) ∨ (𝜓 → 𝜒)) → ((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜒)))
2		idempotor-P4.25b 451	. . . 4 ⊢ ((𝜒 ∨ 𝜒) ↔ 𝜒)
3	2	subimr-P3.40b.RC 328	. . 3 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → 𝜒))
4	3	rcp-NDBIEF0 240	. 2 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ∨ 𝜒)) → ((𝜑 ∧ 𝜓) → 𝜒))
5	1, 4	syl-P3.24.RC 260	1 ⊢ (((𝜑 → 𝜒) ∨ (𝜓 → 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  df-rcp-AND3 161

This theorem is referenced by:  rimoverand-P4.31a  481  rimoverandint-P4.31c  483