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Theorem iinss 3735
 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 2577 . . . 4 𝑦 ∈ V
2 eliin 3689 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 7 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
4 ssel 2966 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
54reximi 2433 . . . 4 (∃𝑥𝐴 𝐵𝐶 → ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
6 r19.36av 2478 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
75, 6syl 14 . . 3 (∃𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 𝑦𝐵𝑦𝐶))
83, 7syl5bi 145 . 2 (∃𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦𝐶))
98ssrdv 2978 1 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324  Vcvv 2574   ⊆ wss 2944  ∩ ciin 3685 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958  df-iin 3687 This theorem is referenced by:  riinm  3756  reliin  4486  cnviinm  4886  iinerm  6208
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