Theorem ididg 3278

Description: A set is identical to itself.

Assertion

Ref	Expression
ididg	|- (A e. B -> AIA)

Proof of Theorem ididg

Step	Hyp	Ref	Expression
1		breq1 2622	. . 3 |- (x = A -> (xIx <-> AIx))
2		breq2 2623	. . 3 |- (x = A -> (AIx <-> AIA))
3	1, 2	bitrd 528	. 2 |- (x = A -> (xIx <-> AIA))
4		eqid 1475	. . 3 |- x = x
5		visset 1813	. . . 4 |- x e. V
6	5	ideq 3277	. . 3 |- (xIx <-> x = x)
7	4, 6	mpbir 190	. 2 |- xIx
8	3, 7	vtoclg 1847	1 |- (A e. B -> AIA)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958   class class class wbr 2619  Icid 2831

This theorem is referenced by:  opelxpex2 3279  fvi 3842

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-id 2835  df-xp 3184  df-rel 3185

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