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Theorem ssiinf 3696
 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 𝐶B)

Proof of Theorem ssiinf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 y V
2 eliin 3652 . . . . 5 (y V → (y x A Bx A y B))
31, 2ax-mp 7 . . . 4 (y x A Bx A y B)
43ralbii 2324 . . 3 (y 𝐶 y x A By 𝐶 x A y B)
5 ssiinf.1 . . . 4 x𝐶
6 nfcv 2175 . . . 4 yA
75, 6ralcomf 2465 . . 3 (y 𝐶 x A y Bx A y 𝐶 y B)
84, 7bitri 173 . 2 (y 𝐶 y x A Bx A y 𝐶 y B)
9 dfss3 2929 . 2 (𝐶 x A By 𝐶 y x A B)
10 dfss3 2929 . . 3 (𝐶By 𝐶 y B)
1110ralbii 2324 . 2 (x A 𝐶Bx A y 𝐶 y B)
128, 9, 113bitr4i 201 1 (𝐶 x A Bx A 𝐶B)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1390  Ⅎwnfc 2162  ∀wral 2300  Vcvv 2551   ⊆ wss 2911  ∩ ciin 3648 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-iin 3650 This theorem is referenced by:  ssiin  3697  dmiin  4522
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