Theorem ereq 4267

Description: Equality theorem for equivalence predicate.

Assertion

Ref	Expression
ereq	|- (R = S -> (Er R <-> Er S))

Proof of Theorem ereq

Step	Hyp	Ref	Expression
1		cnveq 3292	. . . . 5 |- (R = S -> `'R = `'S)
2		coeq1 3281	. . . . . 6 |- (R = S -> (R o. R) = (S o. R))
3		coeq2 3282	. . . . . 6 |- (R = S -> (S o. R) = (S o. S))
4	2, 3	eqtrd 1507	. . . . 5 |- (R = S -> (R o. R) = (S o. S))
5	1, 4	uneq12d 2185	. . . 4 |- (R = S -> (`'R u. (R o. R)) = (`'S u. (S o. S)))
6	5	sseq1d 2088	. . 3 |- (R = S -> ((`'R u. (R o. R)) (_ R <-> (`'S u. (S o. S)) (_ R))
7		sseq2 2083	. . 3 |- (R = S -> ((`'S u. (S o. S)) (_ R <-> (`'S u. (S o. S)) (_ S))
8	6, 7	bitrd 528	. 2 |- (R = S -> ((`'R u. (R o. R)) (_ R <-> (`'S u. (S o. S)) (_ S))
9		df-er 4261	. 2 |- (Er R <-> (`'R u. (R o. R)) (_ R)
10		df-er 4261	. 2 |- (Er S <-> (`'S u. (S o. S)) (_ S)
11	8, 9, 10	3bitr4g 555	1 |- (R = S -> (Er R <-> Er S))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   u. cun 2045   (_ wss 2047  `'ccnv 3169   o. ccom 3174  Er wer 4258

This theorem is referenced by:  erdisj2 10442

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-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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