Theorem dfer2 4262

Description: Alternate definition of equivalence predicate.

Assertion

Ref	Expression
dfer2	|- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))

Distinct variable group:   x,y,z,R

Proof of Theorem dfer2

Step	Hyp	Ref	Expression
1		df-er 4261	. 2 |- (Er R <-> (`'R u. (R o. R)) (_ R)
2		cnvsym 3437	. . . 4 |- (`'R (_ R <-> A.xA.y(xRy -> yRx))
3		cotr 3436	. . . 4 |- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
4	2, 3	anbi12i 482	. . 3 |- ((`'R (_ R /\ (R o. R) (_ R) <-> (A.xA.y(xRy -> yRx) /\ A.xA.yA.z((xRy /\ yRz) -> xRz)))
5		unss 2204	. . 3 |- ((`'R (_ R /\ (R o. R) (_ R) <-> (`'R u. (R o. R)) (_ R)
6		19.28v 1299	. . . . . . 7 |- (A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)) <-> ((xRy -> yRx) /\ A.z((xRy /\ yRz) -> xRz)))
7	6	albii 999	. . . . . 6 |- (A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)) <-> A.y((xRy -> yRx) /\ A.z((xRy /\ yRz) -> xRz)))
8		19.26 1067	. . . . . 6 |- (A.y((xRy -> yRx) /\ A.z((xRy /\ yRz) -> xRz)) <-> (A.y(xRy -> yRx) /\ A.yA.z((xRy /\ yRz) -> xRz)))
9	7, 8	bitr 173	. . . . 5 |- (A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)) <-> (A.y(xRy -> yRx) /\ A.yA.z((xRy /\ yRz) -> xRz)))
10	9	albii 999	. . . 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 1067	. . . 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 174	. . 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 181	. 2 |- ((`'R u. (R o. R)) (_ R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
14	1, 13	bitr 173	1 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   u. cun 2045   (_ wss 2047   class class class wbr 2619  `'ccnv 3169   o. ccom 3174  Er wer 4258

This theorem is referenced by:  ster 4268  ersym 4272  ertr 4274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-co 3187  df-er 4261

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