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Theorem ssiinf 3706
 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 vex 2560 . . . . 5 𝑦 ∈ V
2 eliin 3662 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 7 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
43ralbii 2330 . . 3 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
5 ssiinf.1 . . . 4 𝑥𝐶
6 nfcv 2178 . . . 4 𝑦𝐴
75, 6ralcomf 2471 . . 3 (∀𝑦𝐶𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
84, 7bitri 173 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
9 dfss3 2935 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶 𝑦 𝑥𝐴 𝐵)
10 dfss3 2935 . . 3 (𝐶𝐵 ↔ ∀𝑦𝐶 𝑦𝐵)
1110ralbii 2330 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
128, 9, 113bitr4i 201 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  Ⅎwnfc 2165  ∀wral 2306  Vcvv 2557   ⊆ wss 2917  ∩ ciin 3658 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-iin 3660 This theorem is referenced by:  ssiin  3707  dmiin  4580
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