Metamath Proof Explorer


Theorem caovordd

Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses caovordg.1 ( ( 𝜑 ∧ ( 𝑥𝑆𝑦𝑆𝑧𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
caovordd.2 ( 𝜑𝐴𝑆 )
caovordd.3 ( 𝜑𝐵𝑆 )
caovordd.4 ( 𝜑𝐶𝑆 )
Assertion caovordd ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 caovordg.1 ( ( 𝜑 ∧ ( 𝑥𝑆𝑦𝑆𝑧𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
2 caovordd.2 ( 𝜑𝐴𝑆 )
3 caovordd.3 ( 𝜑𝐵𝑆 )
4 caovordd.4 ( 𝜑𝐶𝑆 )
5 id ( 𝜑𝜑 )
6 1 caovordg ( ( 𝜑 ∧ ( 𝐴𝑆𝐵𝑆𝐶𝑆 ) ) → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
7 5 2 3 4 6 syl13anc ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )