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Theorem ssiinf 4969
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 4915 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3497 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
32ralbii 3162 . . 3 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
4 ssiinf.1 . . . 4 𝑥𝐶
5 nfcv 2974 . . . 4 𝑦𝐴
64, 5ralcomf 3354 . . 3 (∀𝑦𝐶𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
73, 6bitri 276 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
8 dfss3 3953 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶 𝑦 𝑥𝐴 𝐵)
9 dfss3 3953 . . 3 (𝐶𝐵 ↔ ∀𝑦𝐶 𝑦𝐵)
109ralbii 3162 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
117, 8, 103bitr4i 304 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wcel 2105  wnfc 2958  wral 3135  Vcvv 3492  wss 3933   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-in 3940  df-ss 3949  df-iin 4913
This theorem is referenced by:  ssiin  4970  dmiin  5818
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