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Theorem iinss 3699
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 B𝐶 x A 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 2554 . . . 4 y V
2 eliin 3653 . . . 4 (y V → (y x A Bx A y B))
31, 2ax-mp 7 . . 3 (y x A Bx A y B)
4 ssel 2933 . . . . 5 (B𝐶 → (y By 𝐶))
54reximi 2410 . . . 4 (x A B𝐶x A (y By 𝐶))
6 r19.36av 2455 . . . 4 (x A (y By 𝐶) → (x A y By 𝐶))
75, 6syl 14 . . 3 (x A B𝐶 → (x A y By 𝐶))
83, 7syl5bi 141 . 2 (x A B𝐶 → (y x A By 𝐶))
98ssrdv 2945 1 (x A B𝐶 x A B𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  wral 2300  wrex 2301  Vcvv 2551  wss 2911   ciin 3649
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-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iin 3651
This theorem is referenced by:  riinm  3720  reliin  4402  cnviinm  4802  iinerm  6114
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