Theorem intprgOLD 4982

Description: Obsolete version of intprg 4979 as of 1-Sep-2024. (Contributed by FL, 27-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
intprgOLD	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Proof of Theorem intprgOLD

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

Step	Hyp	Ref	Expression
1		preq1 4733	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	inteqd 4949	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑥, 𝑦} = ∩ {𝐴, 𝑦})
3		ineq1 4199	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
4	2, 3	eqeq12d 2741	. 2 ⊢ (𝑥 = 𝐴 → (∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦) ↔ ∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦)))
5		preq2 4734	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
6	5	inteqd 4949	. . 3 ⊢ (𝑦 = 𝐵 → ∩ {𝐴, 𝑦} = ∩ {𝐴, 𝐵})
7		ineq2 4200	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
8	6, 7	eqeq12d 2741	. 2 ⊢ (𝑦 = 𝐵 → (∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦) ↔ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)))
9		vex 3467	. . 3 ⊢ 𝑥 ∈ V
10		vex 3467	. . 3 ⊢ 𝑦 ∈ V
11	9, 10	intpr 4980	. 2 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦)
12	4, 8, 11	vtocl2g 3552	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∩ cin 3938  {cpr 4626  ∩ cint 4944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-un 3944  df-in 3946  df-sn 4625  df-pr 4627  df-int 4945

This theorem is referenced by: (None)