Theorem ereq 4273

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 3298	. . . . 5 |- (R = S -> `'R = `'S)
2		coeq1 3287	. . . . . 6 |- (R = S -> (R o. R) = (S o. R))
3		coeq2 3288	. . . . . 6 |- (R = S -> (S o. R) = (S o. S))
4	2, 3	eqtrd 1510	. . . . 5 |- (R = S -> (R o. R) = (S o. S))
5	1, 4	uneq12d 2188	. . . 4 |- (R = S -> (`'R u. (R o. R)) = (`'S u. (S o. S)))
6	5	sseq1d 2091	. . 3 |- (R = S -> ((`'R u. (R o. R)) (_ R <-> (`'S u. (S o. S)) (_ R))
7		sseq2 2086	. . 3 |- (R = S -> ((`'S u. (S o. S)) (_ R <-> (`'S u. (S o. S)) (_ S))
8	6, 7	bitrd 530	. 2 |- (R = S -> ((`'R u. (R o. R)) (_ R <-> (`'S u. (S o. S)) (_ S))
9		df-er 4267	. 2 |- (Er R <-> (`'R u. (R o. R)) (_ R)
10		df-er 4267	. 2 |- (Er S <-> (`'S u. (S o. S)) (_ S)
11	8, 9, 10	3bitr4g 557	1 |- (R = S -> (Er R <-> Er S))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   u. cun 2048   (_ wss 2050  `'ccnv 3175   o. ccom 3180  Er wer 4264

This theorem is referenced by:  erdisj2 10437

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-cnv 3192  df-co 3193  df-er 4267

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