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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 imbitrrid ⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐷 𝑅 𝐵 ) ) )