MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intprgOLD Structured version   Visualization version   GIF version

Theorem intprgOLD 4988
Description: Obsolete version of intprg 4985 as of 1-Sep-2024. (Contributed by FL, 27-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
intprgOLD ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Proof of Theorem intprgOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 4737 . . . 4 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21inteqd 4955 . . 3 (𝑥 = 𝐴 {𝑥, 𝑦} = {𝐴, 𝑦})
3 ineq1 4205 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
42, 3eqeq12d 2747 . 2 (𝑥 = 𝐴 → ( {𝑥, 𝑦} = (𝑥𝑦) ↔ {𝐴, 𝑦} = (𝐴𝑦)))
5 preq2 4738 . . . 4 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
65inteqd 4955 . . 3 (𝑦 = 𝐵 {𝐴, 𝑦} = {𝐴, 𝐵})
7 ineq2 4206 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
86, 7eqeq12d 2747 . 2 (𝑦 = 𝐵 → ( {𝐴, 𝑦} = (𝐴𝑦) ↔ {𝐴, 𝐵} = (𝐴𝐵)))
9 vex 3477 . . 3 𝑥 ∈ V
10 vex 3477 . . 3 𝑦 ∈ V
119, 10intpr 4986 . 2 {𝑥, 𝑦} = (𝑥𝑦)
124, 8, 11vtocl2g 3563 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  cin 3947  {cpr 4630   cint 4950
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
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  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-un 3953  df-in 3955  df-sn 4629  df-pr 4631  df-int 4951
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator