Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | caovcomg.1 | ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) ) | |
caovcomd.2 | ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) | ||
caovcomd.3 | ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 ) | ||
Assertion | caovcomd | ⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovcomg.1 | ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) ) | |
2 | caovcomd.2 | ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) | |
3 | caovcomd.3 | ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 ) | |
4 | id | ⊢ ( 𝜑 → 𝜑 ) | |
5 | 1 | caovcomg | ⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) ) |
6 | 4 2 3 5 | syl12anc | ⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) ) |