Metamath Proof Explorer

Theorem caovord3d

Description: Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses caovordg.1 ⊢ φ ∧ x ∈ S ∧ y ∈ S ∧ z ∈ S → x R y ↔ z F x R z F y
caovordd.2 ⊢ φ → A ∈ S
caovordd.3 ⊢ φ → B ∈ S
caovordd.4 ⊢ φ → C ∈ S
caovord2d.com ⊢ φ ∧ x ∈ S ∧ y ∈ S → x F y = y F x
caovord3d.5 ⊢ φ → D ∈ S
Assertion caovord3d ⊢ φ → A F B = C F D → A R C ↔ D R B

Proof

Step Hyp Ref Expression
1 caovordg.1 ⊢ φ ∧ x ∈ S ∧ y ∈ S ∧ z ∈ S → x R y ↔ z F x R z F y
2 caovordd.2 ⊢ φ → A ∈ S
3 caovordd.3 ⊢ φ → B ∈ S
4 caovordd.4 ⊢ φ → C ∈ S
5 caovord2d.com ⊢ φ ∧ x ∈ S ∧ y ∈ S → x F y = y F x
6 caovord3d.5 ⊢ φ → D ∈ S
7 breq1 ⊢ A F B = C F D → A F B R C F B ↔ C F D R C F B
8 1 2 4 3 5 caovord2d ⊢ φ → A R C ↔ A F B R C F B
9 1 6 3 4 caovordd ⊢ φ → D R B ↔ C F D R C F B
10 8 9 bibi12d ⊢ φ → A R C ↔ D R B ↔ A F B R C F B ↔ C F D R C F B
11 7 10 syl5ibr ⊢ φ → A F B = C F D → A R C ↔ D R B