Theorem ster 4258

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:   x,y,z,R

Proof of Theorem ster

Step	Hyp	Ref	Expression
1		dfer2 4252	. 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 285	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
5	4	gen2 981	. 2 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
6	1, 5	mpgbir 986	1 |- Er R

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   class class class wbr 2614  Er wer 4248

This theorem is referenced by:  ider 4259  eqer 4261  ecopoprer 4302  ener 4397  hmpher 10459

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-cnv 3181  df-co 3182  df-er 4251

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