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Theorem ssiinf 5059
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1 𝑥𝐶
Assertion
Ref Expression
ssiinf (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)

Proof of Theorem ssiinf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliin 5001 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3483 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
32ralbii 3091 . . 3 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
4 ssiinf.1 . . . 4 𝑥𝐶
5 nfcv 2903 . . . 4 𝑦𝐴
64, 5ralcomf 3300 . . 3 (∀𝑦𝐶𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
73, 6bitri 275 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
8 dfss3 3984 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶 𝑦 𝑥𝐴 𝐵)
9 dfss3 3984 . . 3 (𝐶𝐵 ↔ ∀𝑦𝐶 𝑦𝐵)
109ralbii 3091 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
117, 8, 103bitr4i 303 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2106  wnfc 2888  wral 3059  Vcvv 3478  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:  ssiin  5060  dmiin  5967
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