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Theorem bj-ideqg1ALT 34495
Description: Alternate proof of bj-ideqg1 using brabga 5409 instead of the "unbounded" version bj-brab2a1 34479 or brab2a 5632. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 34487 is in the main section.

Assertion
Ref Expression
bj-ideqg1ALT ((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem bj-ideqg1ALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 5686 . . . 4 Rel I
21brrelex12i 5595 . . 3 (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32adantl 485 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 elex 3498 . . . . 5 (𝐴𝑉𝐴 ∈ V)
54adantr 484 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
6 eleq1 2903 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝑊𝐵𝑊))
76biimparc 483 . . . . 5 ((𝐵𝑊𝐴 = 𝐵) → 𝐴𝑊)
87elexd 3500 . . . 4 ((𝐵𝑊𝐴 = 𝐵) → 𝐴 ∈ V)
95, 8jaoian 954 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
10 eleq1 2903 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
1110biimpac 482 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → 𝐵𝑉)
1211elexd 3500 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → 𝐵 ∈ V)
13 elex 3498 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1413adantr 484 . . . 4 ((𝐵𝑊𝐴 = 𝐵) → 𝐵 ∈ V)
1512, 14jaoian 954 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
169, 15jca 515 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17 eqeq12 2838 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
18 df-id 5448 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1917, 18brabga 5409 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵𝐴 = 𝐵))
203, 16, 19pm5.21nd 801 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  df-rel 5550
This theorem is referenced by: (None)
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