Theorem intprgOLD 4915

Description: Obsolete version of intprg 4912 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 4669	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
2	1	inteqd 4884	. . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑥, 𝑦} = ∩ {𝐴, 𝑦})
3		ineq1 4139	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
4	2, 3	eqeq12d 2754	. 2 ⊢ (𝑥 = 𝐴 → (∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦) ↔ ∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦)))
5		preq2 4670	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
6	5	inteqd 4884	. . 3 ⊢ (𝑦 = 𝐵 → ∩ {𝐴, 𝑦} = ∩ {𝐴, 𝐵})
7		ineq2 4140	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
8	6, 7	eqeq12d 2754	. 2 ⊢ (𝑦 = 𝐵 → (∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦) ↔ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)))
9		vex 3436	. . 3 ⊢ 𝑥 ∈ V
10		vex 3436	. . 3 ⊢ 𝑦 ∈ V
11	9, 10	intpr 4913	. 2 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦)
12	4, 8, 11	vtocl2g 3510	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∩ cin 3886  {cpr 4563  ∩ cint 4879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-un 3892  df-in 3894  df-sn 4562  df-pr 4564  df-int 4880

This theorem is referenced by: (None)