Theorem swopolem 4340

Description: Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)

Hypothesis

Ref	Expression
swopolem.1	|- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( x R y -> ( x R z \/ z R y ) ) )

Assertion

Ref	Expression
swopolem	|- ( ( ph /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( X R Y -> ( X R Z \/ Z R Y ) ) )

Distinct variable groups:   x, y, z, A   ph, x, y, z   x, R, y, z   x, X, y, z   y, Y, z   z, Z

Allowed substitution hints:   Y( x)   Z( x, y)

Proof of Theorem swopolem

Step	Hyp	Ref	Expression
1		swopolem.1	. . 3 |- ( ( ph /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( x R y -> ( x R z \/ z R y ) ) )
2	1	ralrimivvva 2580	. 2 |- ( ph -> A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) )
3		breq1 4036	. . . 4 |- ( x = X -> ( x R y <-> X R y ) )
4		breq1 4036	. . . . 5 |- ( x = X -> ( x R z <-> X R z ) )
5	4	orbi1d 792	. . . 4 |- ( x = X -> ( ( x R z \/ z R y ) <-> ( X R z \/ z R y ) ) )
6	3, 5	imbi12d 234	. . 3 |- ( x = X -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( X R y -> ( X R z \/ z R y ) ) ) )
7		breq2 4037	. . . 4 |- ( y = Y -> ( X R y <-> X R Y ) )
8		breq2 4037	. . . . 5 |- ( y = Y -> ( z R y <-> z R Y ) )
9	8	orbi2d 791	. . . 4 |- ( y = Y -> ( ( X R z \/ z R y ) <-> ( X R z \/ z R Y ) ) )
10	7, 9	imbi12d 234	. . 3 |- ( y = Y -> ( ( X R y -> ( X R z \/ z R y ) ) <-> ( X R Y -> ( X R z \/ z R Y ) ) ) )
11		breq2 4037	. . . . 5 |- ( z = Z -> ( X R z <-> X R Z ) )
12		breq1 4036	. . . . 5 |- ( z = Z -> ( z R Y <-> Z R Y ) )
13	11, 12	orbi12d 794	. . . 4 |- ( z = Z -> ( ( X R z \/ z R Y ) <-> ( X R Z \/ Z R Y ) ) )
14	13	imbi2d 230	. . 3 |- ( z = Z -> ( ( X R Y -> ( X R z \/ z R Y ) ) <-> ( X R Y -> ( X R Z \/ Z R Y ) ) ) )
15	6, 10, 14	rspc3v 2884	. 2 |- ( ( X e. A /\ Y e. A /\ Z e. A ) -> ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) -> ( X R Y -> ( X R Z \/ Z R Y ) ) ) )
16	2, 15	mpan9 281	1 |- ( ( ph /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( X R Y -> ( X R Z \/ Z R Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   class class class wbr 4033

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-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034

This theorem is referenced by:  swoer  6620  swoord1  6621  swoord2  6622