Theorem clifteta 47734

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

Hypotheses

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

Assertion

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

Proof of Theorem clifteta

Step	Hyp	Ref	Expression
1		clifteta.2	. 2 ⊢ 𝜃
2		clifteta.1	. 2 ⊢ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓 ∧ 𝜒))
3	1, 2	2th 267	1 ⊢ (𝜃 ↔ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓 ∧ 𝜒)))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by: (None)