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Theorem iinxprg 3799
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 2157 . . . 4 (𝑥 = 𝐴 → (𝑦𝐶𝑦𝐷))
3 iinxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
43eleq2d 2157 . . . 4 (𝑥 = 𝐵 → (𝑦𝐶𝑦𝐸))
52, 4ralprg 3488 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶 ↔ (𝑦𝐷𝑦𝐸)))
65abbidv 2205 . 2 ((𝐴𝑉𝐵𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶} = {𝑦 ∣ (𝑦𝐷𝑦𝐸)})
7 df-iin 3728 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶}
8 df-in 3003 . 2 (𝐷𝐸) = {𝑦 ∣ (𝑦𝐷𝑦𝐸)}
96, 7, 83eqtr4g 2145 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  {cab 2074  wral 2359  cin 2996  {cpr 3442   ciin 3726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-sn 3447  df-pr 3448  df-iin 3728
This theorem is referenced by: (None)
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