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Theorem iinxprg 4527
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 2668 . . . 4 (𝑥 = 𝐴 → (𝑦𝐶𝑦𝐷))
3 iinxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
43eleq2d 2668 . . . 4 (𝑥 = 𝐵 → (𝑦𝐶𝑦𝐸))
52, 4ralprg 4176 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶 ↔ (𝑦𝐷𝑦𝐸)))
65abbidv 2723 . 2 ((𝐴𝑉𝐵𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶} = {𝑦 ∣ (𝑦𝐷𝑦𝐸)})
7 df-iin 4448 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶}
8 df-in 3542 . 2 (𝐷𝐸) = {𝑦 ∣ (𝑦𝐷𝑦𝐸)}
96, 7, 83eqtr4g 2664 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  {cab 2591  wral 2891  cin 3534  {cpr 4122   ciin 4446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-v 3170  df-sbc 3398  df-un 3540  df-in 3542  df-sn 4121  df-pr 4123  df-iin 4448
This theorem is referenced by:  pmapmeet  33876  diameetN  35162  dihmeetlem2N  35405  dihmeetcN  35408  dihmeet  35449
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