Theorem ideq 4819

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 4818	. 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 2167  Vcvv 2763   class class class wbr 4034   I cid 4324

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671

This theorem is referenced by:  dmi  4882  resieq  4957  resiexg  4992  iss  4993  restidsing  5003  imai  5026  issref  5053  intasym  5055  asymref  5056  intirr  5057  poirr2  5063  cnvi  5075  coi1  5186  idssen  6845