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Theorem ssiin 4970
 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 2975 . 2 𝑥𝐶
21ssiinf 4969 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208  ∀wral 3136   ⊆ wss 3934  ∩ ciin 4911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-v 3495  df-in 3941  df-ss 3950  df-iin 4913 This theorem is referenced by:  triin  5178  cflim2  9677  ptbasfi  22181  limciun  24484  clsint2  33670  fnemeet2  33708  dihglblem4  38425  dihglblem6  38468  iooiinicc  41807  iooiinioc  41821  iinhoiicc  42946  smfsuplem1  43075
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