Theorem papsym 7602

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 4129	. . . 4 |- ( y = Y -> ( X R y <-> X R Y ) )
3		breq1 4128	. . . 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 4128	. . . . . 6 |- ( x = X -> ( x R y <-> X R y ) )
6		breq2 4129	. . . . . 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 2550	. . . 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 7598	. . . . . 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 536	. . . 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 2934	. . 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 2934	. 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 720   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   class class class wbr 4125   X. cxp 4767   Ap wap 7597

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-pap 7598

This theorem is referenced by:  aprlring  14573