Metamath Proof Explorer


Theorem caovord2

Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996)

Ref Expression
Hypotheses caovord.1 𝐴 ∈ V
caovord.2 𝐵 ∈ V
caovord.3 ( 𝑧𝑆 → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
caovord2.3 𝐶 ∈ V
caovord2.com ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 )
Assertion caovord2 ( 𝐶𝑆 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐴 𝐹 𝐶 ) 𝑅 ( 𝐵 𝐹 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 caovord.1 𝐴 ∈ V
2 caovord.2 𝐵 ∈ V
3 caovord.3 ( 𝑧𝑆 → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
4 caovord2.3 𝐶 ∈ V
5 caovord2.com ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 )
6 1 2 3 caovord ( 𝐶𝑆 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
7 4 1 5 caovcom ( 𝐶 𝐹 𝐴 ) = ( 𝐴 𝐹 𝐶 )
8 4 2 5 caovcom ( 𝐶 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐶 )
9 7 8 breq12i ( ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ↔ ( 𝐴 𝐹 𝐶 ) 𝑅 ( 𝐵 𝐹 𝐶 ) )
10 6 9 bitrdi ( 𝐶𝑆 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐴 𝐹 𝐶 ) 𝑅 ( 𝐵 𝐹 𝐶 ) ) )