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Metamath Proof Explorer

Theorem caov31

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995)

Ref Expression
Hypotheses caov.1 ⊢ 𝐴 ∈ V
caov.2 ⊢ 𝐵 ∈ V
caov.3 ⊢ 𝐶 ∈ V
caov.com ⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 )
caov.ass ⊢ ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) )
Assertion caov31 ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 )

Proof

Step Hyp Ref Expression
1 caov.1 ⊢ 𝐴 ∈ V
2 caov.2 ⊢ 𝐵 ∈ V
3 caov.3 ⊢ 𝐶 ∈ V
4 caov.com ⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 )
5 caov.ass ⊢ ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) )
6 1 3 2 5 caovass ⊢ ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) = ( 𝐴 𝐹 ( 𝐶 𝐹 𝐵 ) )
7 1 3 2 4 5 caov12 ⊢ ( 𝐴 𝐹 ( 𝐶 𝐹 𝐵 ) ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
8 6 7 eqtri ⊢ ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
9 1 2 3 4 5 caov32 ⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 )
10 3 1 2 4 5 caov32 ⊢ ( ( 𝐶 𝐹 𝐴 ) 𝐹 𝐵 ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 )
11 3 1 2 5 caovass ⊢ ( ( 𝐶 𝐹 𝐴 ) 𝐹 𝐵 ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
12 10 11 eqtr3i ⊢ ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
13 8 9 12 3eqtr4i ⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 )