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Theorem iinss 3940
Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iinss (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iinss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 2742 . . . 4 𝑦 ∈ V
2 eliin 3893 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 ssel 3151 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
54reximi 2574 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 r19.36av 2628 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
75, 6syl 14 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
83, 7biimtrid 152 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
98ssrdv 3163 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2148  wral 2455  wrex 2456  Vcvv 2739  wss 3131   ciin 3889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iin 3891
This theorem is referenced by:  riinm  3961  reliin  4750  cnviinm  5172  iinerm  6610
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