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Theorem ssiin 5054
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 2904 . 2 𝑥𝐶
21ssiinf 5053 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wral 3060  wss 3950   ciin 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-v 3481  df-ss 3967  df-iin 4993
This theorem is referenced by:  triin  5275  cflim2  10304  ptbasfi  23590  limciun  25930  clsint2  36331  fnemeet2  36369  dihglblem4  41300  dihglblem6  41343  iooiinicc  45560  iooiinioc  45574  iinhoiicc  46694  smfsuplem1  46831
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