This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem brif12

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

Ref Expression
Assertion brif12 ⊢ if φ A B R if φ C D ↔ if- φ A R C B R D

Proof

Step Hyp Ref Expression
1 iftrue ⊢ φ → if φ A B = A
2 iftrue ⊢ φ → if φ C D = C
3 1 2 breq12d ⊢ φ → if φ A B R if φ C D ↔ A R C
4 iffalse ⊢ ¬ φ → if φ A B = B
5 iffalse ⊢ ¬ φ → if φ C D = D
6 4 5 breq12d ⊢ ¬ φ → if φ A B R if φ C D ↔ B R D
7 3 6 casesifp ⊢ if φ A B R if φ C D ↔ if- φ A R C B R D