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Theorem bj-ideqg1ALT 37167
Description: Alternate proof of bj-ideqg1 using brabga 5538 instead of the "unbounded" version bj-brab2a1 37151 or brab2a 5778. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 37159 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 5835 . . . 4 Rel I
21brrelex12i 5739 . . 3 (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32adantl 481 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 elex 3500 . . . . 5 (𝐴𝑉𝐴 ∈ V)
54adantr 480 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
6 eleq1 2828 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝑊𝐵𝑊))
76biimparc 479 . . . . 5 ((𝐵𝑊𝐴 = 𝐵) → 𝐴𝑊)
87elexd 3503 . . . 4 ((𝐵𝑊𝐴 = 𝐵) → 𝐴 ∈ V)
95, 8jaoian 958 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
10 eleq1 2828 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
1110biimpac 478 . . . . 5 ((𝐴𝑉𝐴 = 𝐵) → 𝐵𝑉)
1211elexd 3503 . . . 4 ((𝐴𝑉𝐴 = 𝐵) → 𝐵 ∈ V)
13 elex 3500 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1413adantr 480 . . . 4 ((𝐵𝑊𝐴 = 𝐵) → 𝐵 ∈ V)
1512, 14jaoian 958 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
169, 15jca 511 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
17 eqeq12 2753 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
18 df-id 5577 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1917, 18brabga 5538 . 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 206  wa 395  wo 847   = wceq 1539  wcel 2107  Vcvv 3479   class class class wbr 5142   I cid 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691
This theorem is referenced by: (None)
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