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Theorem ssintab 3623
Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
ssintab (A {xφ} ↔ x(φAx))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem ssintab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssint 3622 . 2 (A {xφ} ↔ y {xφ}Ay)
2 sseq2 2961 . . 3 (y = x → (AyAx))
32ralab2 2699 . 2 (y {xφ}Ayx(φAx))
41, 3bitri 173 1 (A {xφ} ↔ x(φAx))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  {cab 2023  wral 2300  wss 2911   cint 3606
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-int 3607
This theorem is referenced by:  ssmin  3625  ssintrab  3629  intmin4  3634  dfuzi  8104
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