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  4344  opthprc  5680  imadif  6565  frxp2  8074  xrsupss  13205  mdegleb  25994  difrab2  32472  ind1a  32835  poimirlem30  37689  ifpdfan2  43495  ifpdfan  43498  ifpnot  43502  ifpid2  43503  uneqsn  44057  usgrexmpl2nb1  48062  usgrexmpl2nb2  48063  usgrexmpl2nb4  48065