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Metamath Proof Explorer

Theorem caov4

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 ) )
caov.4 ⊢ 𝐷 ∈ V
Assertion caov4 ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) )

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 caov.4 ⊢ 𝐷 ∈ V
7 2 3 6 4 5 caov12 ⊢ ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) )
8 7 oveq2i ⊢ ( 𝐴 𝐹 ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) ) = ( 𝐴 𝐹 ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) ) )
9 ovex ⊢ ( 𝐶 𝐹 𝐷 ) ∈ V
10 1 2 9 5 caovass ⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( 𝐴 𝐹 ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) )
11 ovex ⊢ ( 𝐵 𝐹 𝐷 ) ∈ V
12 1 3 11 5 caovass ⊢ ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) ) = ( 𝐴 𝐹 ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) ) )
13 8 10 12 3eqtr4i ⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) )