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Theorem iinxprg 3887
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 2209 . . . 4 (𝑥 = 𝐴 → (𝑦𝐶𝑦𝐷))
3 iinxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
43eleq2d 2209 . . . 4 (𝑥 = 𝐵 → (𝑦𝐶𝑦𝐸))
52, 4ralprg 3574 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶 ↔ (𝑦𝐷𝑦𝐸)))
65abbidv 2257 . 2 ((𝐴𝑉𝐵𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶} = {𝑦 ∣ (𝑦𝐷𝑦𝐸)})
7 df-iin 3816 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶}
8 df-in 3077 . 2 (𝐷𝐸) = {𝑦 ∣ (𝑦𝐷𝑦𝐸)}
96, 7, 83eqtr4g 2197 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  {cab 2125  wral 2416  cin 3070  {cpr 3528   ciin 3814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-iin 3816
This theorem is referenced by: (None)
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