Theorem bj-ideqgALT 36343

Description: Alternate proof of bj-ideqg 36342 from brabga 5534 instead of bj-opelid 36341 itself proved from bj-opelidb 36337. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ideqgALT	⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem bj-ideqgALT

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reli 5826	. . . 4 ⊢ Rel I
2	1	brrelex12i 5731	. . 3 ⊢ (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	adantl 481	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		bj-inexeqex 36339	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		eqeq12 2748	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
6		df-id 5574	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7	5, 6	brabga 5534	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
8	3, 4, 7	pm5.21nd 799	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ∩ cin 3947   class class class wbr 5148   I cid 5573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683

This theorem is referenced by: (None)