Theorem cliftetb 41105

Description: show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)

Hypotheses

Ref	Expression
cliftetb.1	⊢ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))
cliftetb.2	⊢ 𝜃

Assertion

Ref	Expression
cliftetb	⊢ (𝜃 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))

Proof of Theorem cliftetb

Step	Hyp	Ref	Expression
1		cliftetb.2	. 2 ⊢ 𝜃
2		cliftetb.1	. 2 ⊢ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))
3	1, 2	2th 254	1 ⊢ (𝜃 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197

This theorem is referenced by: (None)