Theorem er2 3201

Description: Alternate definition of equivalence predicate.

Assertion

Ref	Expression
er2	|- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))

Distinct variable group(s):   x,y,z,R

Proof of Theorem er2

Step	Hyp	Ref	Expression
1		df-er 3200	. 2 |- (Er R <-> (`'R u. (R o. R)) (_ R)
2		cnvsym 2626	. . . 4 |- (`'R (_ R <-> A.xA.y(xRy -> yRx))
3		cotr 2625	. . . 4 |- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
4	2, 3	anbi12i 369	. . 3 |- ((`'R (_ R /\ (R o. R) (_ R) <-> (A.xA.y(xRy -> yRx) /\ A.xA.yA.z((xRy /\ yRz) -> xRz)))
5		unss 1632	. . 3 |- ((`'R (_ R /\ (R o. R) (_ R) <-> (`'R u. (R o. R)) (_ R)
6		19.28v 957	. . . . . . 7 |- (A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)) <-> ((xRy -> yRx) /\ A.z((xRy /\ yRz) -> xRz)))
7	6	bial 695	. . . . . 6 |- (A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)) <-> A.y((xRy -> yRx) /\ A.z((xRy /\ yRz) -> xRz)))
8		19.26 749	. . . . . 6 |- (A.y((xRy -> yRx) /\ A.z((xRy /\ yRz) -> xRz)) <-> (A.y(xRy -> yRx) /\ A.yA.z((xRy /\ yRz) -> xRz)))
9	7, 8	bitr 151	. . . . 5 |- (A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)) <-> (A.y(xRy -> yRx) /\ A.yA.z((xRy /\ yRz) -> xRz)))
10	9	bial 695	. . . 4 |- (A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)) <-> A.x(A.y(xRy -> yRx) /\ A.yA.z((xRy /\ yRz) -> xRz)))
11		19.26 749	. . . 4 |- (A.x(A.y(xRy -> yRx) /\ A.yA.z((xRy /\ yRz) -> xRz)) <-> (A.xA.y(xRy -> yRx) /\ A.xA.yA.z((xRy /\ yRz) -> xRz)))
12	10, 11	bitr2 152	. . 3 |- ((A.xA.y(xRy -> yRx) /\ A.xA.yA.z((xRy /\ yRz) -> xRz)) <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
13	4, 5, 12	3bitr3 156	. 2 |- ((`'R u. (R o. R)) (_ R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
14	1, 13	bitr 151	1 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   u. cun 1485   (_ wss 1487   class class class wbr 2054  `'ccnv 2409   o. ccom 2414  Er wer 3197

This theorem is referenced by:  ster 3207  ersym 3209  ertr 3211

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-co 2427  df-er 3200

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