Theorem bj-ideqgALT 37470

Description: Alternate proof of bj-ideqg 37469 from brabga 5480 instead of bj-opelid 37468 itself proved from bj-opelidb 37464. (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 5773	. . . 4 ⊢ Rel I
2	1	brrelex12i 5677	. . 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 37466	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		eqeq12 2754	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
6		df-id 5517	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
7	5, 6	brabga 5480	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
8	3, 4, 7	pm5.21nd 802	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889   class class class wbr 5086   I cid 5516

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629

This theorem is referenced by: (None)