Theorem biorfri 939

Description: A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.) (Proof shortened by AV, 10-Aug-2025.)

Hypothesis

Ref	Expression
biorfi.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
biorfri	⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))

Proof of Theorem biorfri

Step	Hyp	Ref	Expression
1		biorfi.1	. . 3 ⊢ ¬ 𝜑
2	1	biorfi 938	. 2 ⊢ (𝜓 ↔ (𝜑 ∨ 𝜓))
3		orcom 870	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
4	2, 3	bitri 275	1 ⊢ (𝜓 ↔ (𝜓 ∨ 𝜑))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 847

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 207  df-or 848

This theorem is referenced by:  pm4.43  1024  dn1  1057  un0  4343  opthprc  5683  imadif  6570  frxp2  8080  xrsupss  13210  mdegleb  25997  difrab2  32479  ind1a  32845  poimirlem30  37710  ifpdfan2  43580  ifpdfan  43583  ifpnot  43587  ifpid2  43588  uneqsn  44142  usgrexmpl2nb1  48156  usgrexmpl2nb2  48157  usgrexmpl2nb4  48159