Theorem caovordig 6135

Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)

Hypothesis

Ref	Expression
caovordig.1	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y -> ( z F x ) R ( z F y ) ) )

Assertion

Ref	Expression
caovordig	|- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B -> ( C F A ) R ( C F B ) ) )

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

Proof of Theorem caovordig

Step	Hyp	Ref	Expression
1		caovordig.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( x R y -> ( z F x ) R ( z F y ) ) )
2	1	ralrimivvva 2591	. 2 |- ( ph -> A. x e. S A. y e. S A. z e. S ( x R y -> ( z F x ) R ( z F y ) ) )
3		breq1 4062	. . . 4 |- ( x = A -> ( x R y <-> A R y ) )
4		oveq2 5975	. . . . 5 |- ( x = A -> ( z F x ) = ( z F A ) )
5	4	breq1d 4069	. . . 4 |- ( x = A -> ( ( z F x ) R ( z F y ) <-> ( z F A ) R ( z F y ) ) )
6	3, 5	imbi12d 234	. . 3 |- ( x = A -> ( ( x R y -> ( z F x ) R ( z F y ) ) <-> ( A R y -> ( z F A ) R ( z F y ) ) ) )
7		breq2 4063	. . . 4 |- ( y = B -> ( A R y <-> A R B ) )
8		oveq2 5975	. . . . 5 |- ( y = B -> ( z F y ) = ( z F B ) )
9	8	breq2d 4071	. . . 4 |- ( y = B -> ( ( z F A ) R ( z F y ) <-> ( z F A ) R ( z F B ) ) )
10	7, 9	imbi12d 234	. . 3 |- ( y = B -> ( ( A R y -> ( z F A ) R ( z F y ) ) <-> ( A R B -> ( z F A ) R ( z F B ) ) ) )
11		oveq1 5974	. . . . 5 |- ( z = C -> ( z F A ) = ( C F A ) )
12		oveq1 5974	. . . . 5 |- ( z = C -> ( z F B ) = ( C F B ) )
13	11, 12	breq12d 4072	. . . 4 |- ( z = C -> ( ( z F A ) R ( z F B ) <-> ( C F A ) R ( C F B ) ) )
14	13	imbi2d 230	. . 3 |- ( z = C -> ( ( A R B -> ( z F A ) R ( z F B ) ) <-> ( A R B -> ( C F A ) R ( C F B ) ) ) )
15	6, 10, 14	rspc3v 2900	. 2 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A. x e. S A. y e. S A. z e. S ( x R y -> ( z F x ) R ( z F y ) ) -> ( A R B -> ( C F A ) R ( C F B ) ) ) )
16	2, 15	mpan9 281	1 |- ( ( ph /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B -> ( C F A ) R ( C F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   class class class wbr 4059  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  caovordid  6136