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Theorem iinxprg 4753
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 2825 . . . 4 (𝑥 = 𝐴 → (𝑦𝐶𝑦𝐷))
3 iinxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
43eleq2d 2825 . . . 4 (𝑥 = 𝐵 → (𝑦𝐶𝑦𝐸))
52, 4ralprg 4378 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶 ↔ (𝑦𝐷𝑦𝐸)))
65abbidv 2879 . 2 ((𝐴𝑉𝐵𝑊) → {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶} = {𝑦 ∣ (𝑦𝐷𝑦𝐸)})
7 df-iin 4675 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = {𝑦 ∣ ∀𝑥 ∈ {𝐴, 𝐵}𝑦𝐶}
8 df-in 3722 . 2 (𝐷𝐸) = {𝑦 ∣ (𝑦𝐷𝑦𝐸)}
96, 7, 83eqtr4g 2819 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  cin 3714  {cpr 4323   ciin 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-sbc 3577  df-un 3720  df-in 3722  df-sn 4322  df-pr 4324  df-iin 4675
This theorem is referenced by:  pmapmeet  35580  diameetN  36865  dihmeetlem2N  37108  dihmeetcN  37111  dihmeet  37152
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