Theorem idornthmr-P4.24b 449

Description: '∨' Identity is Any Refuted Statement (right). †

Hypothesis

Ref	Expression
idornthmr-P4.24b.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
idornthmr-P4.24b	⊢ ((𝜓 ∨ 𝜑) ↔ 𝜓)

Proof of Theorem idornthmr-P4.24b

Step	Hyp	Ref	Expression
1		idorfalser-P4.20b 441	. 2 ⊢ ((𝜓 ∨ ⊥) ↔ 𝜓)
2		idornthmr-P4.24b.1	. . . . . 6 ⊢ ¬ 𝜑
3	2	nthmeqfalse-P4.21b 443	. . . . 5 ⊢ (𝜑 ↔ ⊥)
4	3	suborr-P3.43b.RC 349	. . . 4 ⊢ ((𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ⊥))
5	4	subbil-P3.41a.RC 333	. . 3 ⊢ (((𝜓 ∨ 𝜑) ↔ 𝜓) ↔ ((𝜓 ∨ ⊥) ↔ 𝜓))
6	5	rcp-NDBIER0 241	. 2 ⊢ (((𝜓 ∨ ⊥) ↔ 𝜓) → ((𝜓 ∨ 𝜑) ↔ 𝜓))
7	1, 6	rcp-NDIME0 228	1 ⊢ ((𝜓 ∨ 𝜑) ↔ 𝜓)

Syntax hints:  ¬ wff-neg 9   ↔ wff-bi 104   ∨ wff-or 144  ⊥wff-false 157

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-false-D2.5 158  df-rcp-AND3 161

This theorem is referenced by:  biasandor-P4.34a  491