Theorem ster 3207

Description: A symmetric, transitive relation is an equivalence relation.

Hypotheses

Ref	Expression
ster.1	|- (xRy -> yRx)
ster.2	|- ((xRy /\ yRz) -> xRz)

Assertion

Ref	Expression
ster	|- Er R

Distinct variable group(s):   x,y,z,R

Proof of Theorem ster

Step	Hyp	Ref	Expression
1		er2 3201	. 2 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
2		ster.1	. . . 4 |- (xRy -> yRx)
3		ster.2	. . . 4 |- ((xRy /\ yRz) -> xRz)
4	2, 3	pm3.2i 234	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
5	4	gen2 681	. 2 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
6	1, 5	mpgbir 686	1 |- Er R

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   class class class wbr 2054  Er wer 3197

This theorem is referenced by:  ider 3208  ecopoprer 3248  ener 3313

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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