Metamath Proof Explorer


Theorem caovcomd

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

Ref Expression
Hypotheses caovcomg.1 ( ( 𝜑 ∧ ( 𝑥𝑆𝑦𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
caovcomd.2 ( 𝜑𝐴𝑆 )
caovcomd.3 ( 𝜑𝐵𝑆 )
Assertion caovcomd ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )

Proof

Step Hyp Ref Expression
1 caovcomg.1 ( ( 𝜑 ∧ ( 𝑥𝑆𝑦𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
2 caovcomd.2 ( 𝜑𝐴𝑆 )
3 caovcomd.3 ( 𝜑𝐵𝑆 )
4 id ( 𝜑𝜑 )
5 1 caovcomg ( ( 𝜑 ∧ ( 𝐴𝑆𝐵𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )
6 4 2 3 5 syl12anc ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )