Theorem caov32 6111

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

Hypotheses

Ref	Expression
caov.1	|- A e. _V
caov.2	|- B e. _V
caov.3	|- C e. _V
caov.com	|- ( x F y ) = ( y F x )
caov.ass	|- ( ( x F y ) F z ) = ( x F ( y F z ) )

Assertion

Ref	Expression
caov32	|- ( ( A F B ) F C ) = ( ( A F C ) F B )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, F, y, z

Proof of Theorem caov32

Step	Hyp	Ref	Expression
1		caov.2	. . . 4 |- B e. _V
2		caov.3	. . . 4 |- C e. _V
3		caov.com	. . . 4 |- ( x F y ) = ( y F x )
4	1, 2, 3	caovcom 6081	. . 3 |- ( B F C ) = ( C F B )
5	4	oveq2i 5933	. 2 |- ( A F ( B F C ) ) = ( A F ( C F B ) )
6		caov.1	. . 3 |- A e. _V
7		caov.ass	. . 3 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
8	6, 1, 2, 7	caovass 6084	. 2 |- ( ( A F B ) F C ) = ( A F ( B F C ) )
9	6, 2, 1, 7	caovass 6084	. 2 |- ( ( A F C ) F B ) = ( A F ( C F B ) )
10	5, 8, 9	3eqtr4i 2227	1 |- ( ( A F B ) F C ) = ( ( A F C ) F B )

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763  (class class class)co 5922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  caov31  6113