Theorem ster 4268

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 4262	. 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 983	. 2 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
6	1, 5	mpgbir 988	1 |- Er R

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   class class class wbr 2619  Er wer 4258

This theorem is referenced by:  ider 4269  eqer 4271  ecopoprer 4312  ener 4410  hmpher 10536

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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