Theorem biortn 745

Description: A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)

Assertion

Ref	Expression
biortn	⊢ (𝜑 → (𝜓 ↔ (¬ 𝜑 ∨ 𝜓)))

Proof of Theorem biortn

Step	Hyp	Ref	Expression
1		notnot 629	. 2 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		biorf 744	. 2 ⊢ (¬ ¬ 𝜑 → (𝜓 ↔ (¬ 𝜑 ∨ 𝜓)))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝜓 ↔ (¬ 𝜑 ∨ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  oranabs  815