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Theorem iinxprg 4537
 Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
iinxprg.1 (𝑥 = 𝐴𝐶 = 𝐷)
iinxprg.2 (𝑥 = 𝐵𝐶 = 𝐸)
Assertion
Ref Expression
iinxprg ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸
Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iinxprg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iinxprg.1 . . . . 5 (𝑥 = 𝐴𝐶 = 𝐷)
21eleq2d 2673 . . . 4 (𝑥 = 𝐴 → (𝑦𝐶𝑦𝐷))
3 iinxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
43eleq2d 2673 . . . 4 (𝑥 = 𝐵 → (𝑦𝐶𝑦𝐸))
52, 4ralprg 4181 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶 ↔ (𝑦𝐷𝑦𝐸)))
65abbidv 2728 . 2 ((𝐴𝑉𝐵𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶} = {𝑦 ∣ (𝑦𝐷𝑦𝐸)})
7 df-iin 4458 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶}
8 df-in 3547 . 2 (𝐷𝐸) = {𝑦 ∣ (𝑦𝐷𝑦𝐸)}
96, 7, 83eqtr4g 2669 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896   ∩ cin 3539  {cpr 4127  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-sbc 3403  df-un 3545  df-in 3547  df-sn 4126  df-pr 4128  df-iin 4458 This theorem is referenced by:  pmapmeet  34077  diameetN  35363  dihmeetlem2N  35606  dihmeetcN  35609  dihmeet  35650
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