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Theorem bj-ideqg1 34494
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 34487 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 2838 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
2 df-id 5448 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
31, 2bj-brab2a1 34479 . 2 (𝐴 I 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
4 simpr 488 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
5 elex 3498 . . . . . . 7 (𝐴𝑉𝐴 ∈ V)
65a1d 25 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵𝐴 ∈ V))
7 elex 3498 . . . . . . 7 (𝐵𝑊𝐵 ∈ V)
8 eleq1a 2911 . . . . . . 7 (𝐵 ∈ V → (𝐴 = 𝐵𝐴 ∈ V))
97, 8syl 17 . . . . . 6 (𝐵𝑊 → (𝐴 = 𝐵𝐴 ∈ V))
106, 9jaoi 854 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵𝐴 ∈ V))
11 eleq1 2903 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
125, 11syl5ibcom 248 . . . . . 6 (𝐴𝑉 → (𝐴 = 𝐵𝐵 ∈ V))
137a1d 25 . . . . . 6 (𝐵𝑊 → (𝐴 = 𝐵𝐵 ∈ V))
1412, 13jaoi 854 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵𝐵 ∈ V))
1510, 14jcad 516 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
1615ancrd 555 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)))
174, 16impbid2 229 . 2 ((𝐴𝑉𝐵𝑊) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵))
183, 17syl5bb 286 1 ((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  Vcvv 3480   class class class wbr 5053   I cid 5447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549
This theorem is referenced by: (None)
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