Theorem biorfi 421

Description: A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.)

Hypothesis

Ref	Expression
biorfi.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
biorfi	⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))

Proof of Theorem biorfi

Step	Hyp	Ref	Expression
1		orc 399	. 2 ⊢ (𝜓 → (𝜓 ∨ 𝜑))
2		biorfi.1	. . 3 ⊢ ¬ 𝜑
3		orel2 397	. . 3 ⊢ (¬ 𝜑 → ((𝜓 ∨ 𝜑) → 𝜓))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝜓 ∨ 𝜑) → 𝜓)
5	1, 4	impbii 198	1 ⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384

This theorem is referenced by:  pm4.43  964  dn1  1000  indifdir  3842  un0  3919  opthprc  5089  imadif  5887  xrsupss  12011  mdegleb  23628  ind1a  29410  poimirlem30  32609  ifpdfan2  36826  ifpdfan  36829  ifpnot  36833  ifpid2  36834  uneqsn  37341