Theorem ideq 4815

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)

Hypothesis

Ref	Expression
ideq.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
ideq	⊢ (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)

Proof of Theorem ideq

Step	Hyp	Ref	Expression
1		ideq.1	. 2 ⊢ 𝐵 ∈ V
2		ideqg 4814	. 2 ⊢ (𝐵 ∈ V → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  Vcvv 2760   class class class wbr 4030   I cid 4320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667

This theorem is referenced by:  dmi  4878  resieq  4953  resiexg  4988  iss  4989  restidsing  4999  imai  5022  issref  5049  intasym  5051  asymref  5052  intirr  5053  poirr2  5059  cnvi  5071  coi1  5182  idssen  6833