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Theorem bj-ideqg1ALT 35344
Description: Alternate proof of bj-ideqg1 using brabga 5444 instead of the "unbounded" version bj-brab2a1 35328 or brab2a 5674. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 35336 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 5729 . . . 4 Rel I
21brrelex12i 5637 . . 3 (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32adantl 482 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 elex 3447 . . . . 5 (𝐴𝑉𝐴 ∈ V)
54adantr 481 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
6 eleq1 2826 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝑊𝐵𝑊))
76biimparc 480 . . . . 5 ((𝐵𝑊𝐴 = 𝐵) → 𝐴𝑊)
87elexd 3449 . . . 4 ((𝐵𝑊𝐴 = 𝐵) → 𝐴 ∈ V)
95, 8jaoian 954 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
10 eleq1 2826 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
1110biimpac 479 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → 𝐵𝑉)
1211elexd 3449 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → 𝐵 ∈ V)
13 elex 3447 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1413adantr 481 . . . 4 ((𝐵𝑊𝐴 = 𝐵) → 𝐵 ∈ V)
1512, 14jaoian 954 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
169, 15jca 512 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17 eqeq12 2755 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
18 df-id 5484 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1917, 18brabga 5444 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵𝐴 = 𝐵))
203, 16, 19pm5.21nd 799 1 ((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  Vcvv 3429   class class class wbr 5073   I cid 5483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5074  df-opab 5136  df-id 5484  df-xp 5590  df-rel 5591
This theorem is referenced by: (None)
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