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Theorem iinss 4017
 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 (x A B Cx A B C)
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 2862 . . . 4 y V
2 eliin 3974 . . . 4 (y V → (y x A Bx A y B))
31, 2ax-mp 5 . . 3 (y x A Bx A y B)
4 ssel 3267 . . . . 5 (B C → (y By C))
54reximi 2721 . . . 4 (x A B Cx A (y By C))
6 r19.36av 2759 . . . 4 (x A (y By C) → (x A y By C))
75, 6syl 15 . . 3 (x A B C → (x A y By C))
83, 7syl5bi 208 . 2 (x A B C → (y x A By C))
98ssrdv 3278 1 (x A B Cx A B C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ∩ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iin 3972 This theorem is referenced by:  riinn0  4040
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