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Theorem intprgOLD 4950
Description: Obsolete version of intprg 4947 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 4699 . . . 4 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21inteqd 4917 . . 3 (𝑥 = 𝐴 {𝑥, 𝑦} = {𝐴, 𝑦})
3 ineq1 4170 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
42, 3eqeq12d 2753 . 2 (𝑥 = 𝐴 → ( {𝑥, 𝑦} = (𝑥𝑦) ↔ {𝐴, 𝑦} = (𝐴𝑦)))
5 preq2 4700 . . . 4 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
65inteqd 4917 . . 3 (𝑦 = 𝐵 {𝐴, 𝑦} = {𝐴, 𝐵})
7 ineq2 4171 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
86, 7eqeq12d 2753 . 2 (𝑦 = 𝐵 → ( {𝐴, 𝑦} = (𝐴𝑦) ↔ {𝐴, 𝐵} = (𝐴𝐵)))
9 vex 3452 . . 3 𝑥 ∈ V
10 vex 3452 . . 3 𝑦 ∈ V
119, 10intpr 4948 . 2 {𝑥, 𝑦} = (𝑥𝑦)
124, 8, 11vtocl2g 3534 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cin 3914  {cpr 4593   cint 4912
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-un 3920  df-in 3922  df-sn 4592  df-pr 4594  df-int 4913
This theorem is referenced by: (None)
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