This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem caovcanrd

Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses caovcang.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) )
caovcand.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑇 )
caovcand.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
caovcand.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
caovcanrd.5 ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
caovcanrd.6 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
Assertion caovcanrd ( 𝜑 → ( ( 𝐵 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) ↔ 𝐵 = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 caovcang.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) )
2 caovcand.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑇 )
3 caovcand.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4 caovcand.4 ⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
5 caovcanrd.5 ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
6 caovcanrd.6 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
7 6 5 3 caovcomd ⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )
8 6 5 4 caovcomd ⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐶 𝐹 𝐴 ) )
9 7 8 eqeq12d ⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ ( 𝐵 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) ) )
10 1 2 3 4 caovcand ⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 𝐵 = 𝐶 ) )
11 9 10 bitr3d ⊢ ( 𝜑 → ( ( 𝐵 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) ↔ 𝐵 = 𝐶 ) )