Theorem cliftet 43179

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

Hypotheses

Ref	Expression
cliftet.1	⊢ (𝜑 ∧ 𝜒)
cliftet.2	⊢ 𝜃

Assertion

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

Proof of Theorem cliftet

Step	Hyp	Ref	Expression
1		cliftet.2	. 2 ⊢ 𝜃
2		cliftet.1	. . 3 ⊢ (𝜑 ∧ 𝜒)
3	2	orci 861	. 2 ⊢ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))
4	1, 3	2th 266	1 ⊢ (𝜃 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843

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 209  df-or 844

This theorem is referenced by: (None)