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Theorem bj-ideqg1 36700
Description: For sets, the identity relation is the same thing as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ, 24-Dec-2023.)

TODO: delete once bj-ideqg 36693 is in the main section.

Assertion
Ref Expression
bj-ideqg1 ((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem bj-ideqg1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq12 2742 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
2 df-id 5570 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
31, 2bj-brab2a1 36685 . 2 (𝐴 I 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
4 simpr 483 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
5 elex 3482 . . . . . . 7 (𝐴𝑉𝐴 ∈ V)
65a1d 25 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵𝐴 ∈ V))
7 elex 3482 . . . . . . 7 (𝐵𝑊𝐵 ∈ V)
8 eleq1a 2820 . . . . . . 7 (𝐵 ∈ V → (𝐴 = 𝐵𝐴 ∈ V))
97, 8syl 17 . . . . . 6 (𝐵𝑊 → (𝐴 = 𝐵𝐴 ∈ V))
106, 9jaoi 855 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵𝐴 ∈ V))
11 eleq1 2813 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
125, 11syl5ibcom 244 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵𝐵 ∈ V))
137a1d 25 . . . . . 6 (𝐵𝑊 → (𝐴 = 𝐵𝐵 ∈ V))
1412, 13jaoi 855 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵𝐵 ∈ V))
1510, 14jcad 511 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
1615ancrd 550 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)))
174, 16impbid2 225 . 2 ((𝐴𝑉𝐵𝑊) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵))
183, 17bitrid 282 1 ((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  Vcvv 3463   class class class wbr 5143   I cid 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678
This theorem is referenced by: (None)
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