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Theorem ssiin 4791
Description: Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
ssiin (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ssiin
StepHypRef Expression
1 nfcv 2970 . 2 𝑥𝐶
21ssiinf 4790 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wral 3118  wss 3799   ciin 4742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-v 3417  df-in 3806  df-ss 3813  df-iin 4744
This theorem is referenced by:  triin  4991  cflim2  9401  ptbasfi  21756  limciun  24058  clsint2  32863  fnemeet2  32901  dihglblem4  37373  dihglblem6  37416  iooiinicc  40565  iooiinioc  40579  iinhoiicc  41683  smfsuplem1  41812
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