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Theorem ssiinf 4015
 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 x A Bx A C B)

Proof of Theorem ssiinf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . 5 y V
2 eliin 3974 . . . . 5 (y V → (y x A Bx A y B))
31, 2ax-mp 5 . . . 4 (y x A Bx A y B)
43ralbii 2638 . . 3 (y C y x A By C x A y B)
5 ssiinf.1 . . . 4 xC
6 nfcv 2489 . . . 4 yA
75, 6ralcomf 2769 . . 3 (y C x A y Bx A y C y B)
84, 7bitri 240 . 2 (y C y x A Bx A y C y B)
9 dfss3 3263 . 2 (C x A By C y x A B)
10 dfss3 3263 . . 3 (C By C y B)
1110ralbii 2638 . 2 (x A C Bx A y C y B)
128, 9, 113bitr4i 268 1 (C x A Bx A C B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iin 3972 This theorem is referenced by:  ssiin  4016  dmiin  4965
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