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Theorem ssiinf 4944
 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 4889 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3449 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
32ralbii 3136 . . 3 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
4 ssiinf.1 . . . 4 𝑥𝐶
5 nfcv 2958 . . . 4 𝑦𝐴
64, 5ralcomf 3313 . . 3 (∀𝑦𝐶𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
73, 6bitri 278 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
8 dfss3 3906 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶 𝑦 𝑥𝐴 𝐵)
9 dfss3 3906 . . 3 (𝐶𝐵 ↔ ∀𝑦𝐶 𝑦𝐵)
109ralbii 3136 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
117, 8, 103bitr4i 306 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2112  Ⅎwnfc 2939  ∀wral 3109  Vcvv 3444   ⊆ wss 3884  ∩ ciin 4885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-in 3891  df-ss 3901  df-iin 4887 This theorem is referenced by:  ssiin  4945  dmiin  5793
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