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Theorem ssiin 4702
 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 2912 . 2 𝑥𝐶
21ssiinf 4701 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∀wral 3060   ⊆ wss 3721  ∩ ciin 4653 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-v 3351  df-in 3728  df-ss 3735  df-iin 4655 This theorem is referenced by:  cflim2  9286  ptbasfi  21604  limciun  23877  clsint2  32655  fnemeet2  32693  dihglblem4  37100  dihglblem6  37143  iooiinicc  40281  iooiinioc  40295  iinhoiicc  41402  smfsuplem1  41531
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