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Metamath Proof Explorer

Theorem brif1

Description: Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024)

Ref Expression
Assertion brif1 ⊢ if φ A B R C ↔ if- φ A R C B R C

Proof

Step Hyp Ref Expression
1 iftrue ⊢ φ → if φ A B = A
2 1 breq1d ⊢ φ → if φ A B R C ↔ A R C
3 iffalse ⊢ ¬ φ → if φ A B = B
4 3 breq1d ⊢ ¬ φ → if φ A B R C ↔ B R C
5 2 4 casesifp ⊢ if φ A B R C ↔ if- φ A R C B R C