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Theorem ssiin 5060
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 2903 . 2 𝑥𝐶
21ssiinf 5059 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wral 3059  wss 3963   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-v 3480  df-ss 3980  df-iin 4999
This theorem is referenced by:  triin  5282  cflim2  10301  ptbasfi  23605  limciun  25944  clsint2  36312  fnemeet2  36350  dihglblem4  41280  dihglblem6  41323  iooiinicc  45495  iooiinioc  45509  iinhoiicc  46630  smfsuplem1  46767
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