Theorem papsym 7576

Description: An apartness is symmetric. (Contributed by Jim Kingdon, 27-May-2026.)

Hypotheses

Ref	Expression
papsym.r	|- ( ph -> R Ap A )
papsym.x	|- ( ph -> X e. A )
papsym.y	|- ( ph -> Y e. A )
papsym.ap	|- ( ph -> X R Y )

Assertion

Ref	Expression
papsym	|- ( ph -> Y R X )

Proof of Theorem papsym

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		papsym.ap	. 2 |- ( ph -> X R Y )
2		breq2 4118	. . . 4 |- ( y = Y -> ( X R y <-> X R Y ) )
3		breq1 4117	. . . 4 |- ( y = Y -> ( y R X <-> Y R X ) )
4	2, 3	imbi12d 234	. . 3 |- ( y = Y -> ( ( X R y -> y R X ) <-> ( X R Y -> Y R X ) ) )
5		breq1 4117	. . . . . 6 |- ( x = X -> ( x R y <-> X R y ) )
6		breq2 4118	. . . . . 6 |- ( x = X -> ( y R x <-> y R X ) )
7	5, 6	imbi12d 234	. . . . 5 |- ( x = X -> ( ( x R y -> y R x ) <-> ( X R y -> y R X ) ) )
8	7	ralbidv 2544	. . . 4 |- ( x = X -> ( A. y e. A ( x R y -> y R x ) <-> A. y e. A ( X R y -> y R X ) ) )
9		papsym.r	. . . . . 6 |- ( ph -> R Ap A )
10		df-pap 7572	. . . . . 6 |- ( R Ap A <-> ( ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) /\ ( A. x e. A A. y e. A ( x R y -> y R x ) /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) ) ) )
11	9, 10	sylib 122	. . . . 5 |- ( ph -> ( ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) /\ ( A. x e. A A. y e. A ( x R y -> y R x ) /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) ) ) )
12	11	simprld 532	. . . 4 |- ( ph -> A. x e. A A. y e. A ( x R y -> y R x ) )
13		papsym.x	. . . 4 |- ( ph -> X e. A )
14	8, 12, 13	rspcdva 2928	. . 3 |- ( ph -> A. y e. A ( X R y -> y R X ) )
15		papsym.y	. . 3 |- ( ph -> Y e. A )
16	4, 14, 15	rspcdva 2928	. 2 |- ( ph -> ( X R Y -> Y R X ) )
17	1, 16	mpd 13	1 |- ( ph -> Y R X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   A.wral 2522   C_ wss 3214   class class class wbr 4114   X. cxp 4752   Ap wap 7571

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-pap 7572

This theorem is referenced by:  aprlring  14523