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Theorem bj-ideqgALT 37140
Description: Alternate proof of bj-ideqg 37139 from brabga 5543 instead of bj-opelid 37138 itself proved from bj-opelidb 37134. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ideqgALT ((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem bj-ideqgALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 5838 . . . 4 Rel I
21brrelex12i 5743 . . 3 (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32adantl 481 . 2 (((𝐴𝐵) ∈ 𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 bj-inexeqex 37136 . 2 (((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 eqeq12 2751 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
6 df-id 5582 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
75, 6brabga 5543 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵𝐴 = 𝐵))
83, 4, 7pm5.21nd 802 1 ((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961   class class class wbr 5147   I cid 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695
This theorem is referenced by: (None)
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