Theorem biorfi 396

Description: A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)

Hypothesis

Ref	Expression
biorfi.1	⊢ ¬ φ

Assertion

Ref	Expression
biorfi	⊢ (ψ ↔ (ψ ∨ φ))

Proof of Theorem biorfi

Step	Hyp	Ref	Expression
1		biorfi.1	. 2 ⊢ ¬ φ
2		orc 374	. . 3 ⊢ (ψ → (ψ ∨ φ))
3		orel2 372	. . 3 ⊢ (¬ φ → ((ψ ∨ φ) → ψ))
4	2, 3	impbid2 195	. 2 ⊢ (¬ φ → (ψ ↔ (ψ ∨ φ)))
5	1, 4	ax-mp 5	1 ⊢ (ψ ↔ (ψ ∨ φ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357

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 177  df-or 359

This theorem is referenced by:  pm4.43  893  dn1  932  indifdir  3512  un0  3576  eqtfinrelk  4487  proj1op  4601  proj2op  4602  imadif  5172