Metamath Proof Explorer


Theorem caovcld

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

Ref Expression
Hypotheses caovclg.1 ( ( 𝜑 ∧ ( 𝑥𝐶𝑦𝐷 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝐸 )
caovcld.2 ( 𝜑𝐴𝐶 )
caovcld.3 ( 𝜑𝐵𝐷 )
Assertion caovcld ( 𝜑 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 )

Proof

Step Hyp Ref Expression
1 caovclg.1 ( ( 𝜑 ∧ ( 𝑥𝐶𝑦𝐷 ) ) → ( 𝑥 𝐹 𝑦 ) ∈ 𝐸 )
2 caovcld.2 ( 𝜑𝐴𝐶 )
3 caovcld.3 ( 𝜑𝐵𝐷 )
4 id ( 𝜑𝜑 )
5 1 caovclg ( ( 𝜑 ∧ ( 𝐴𝐶𝐵𝐷 ) ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 )
6 4 2 3 5 syl12anc ( 𝜑 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐸 )