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Theorem ssintab 3943
 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 {x φ} ↔ x(φA x))
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 3942 . 2 (A {x φ} ↔ y {x φ}A y)
2 sseq2 3293 . . 3 (y = x → (A yA x))
32ralab2 3001 . 2 (y {x φ}A yx(φA x))
41, 3bitri 240 1 (A {x φ} ↔ x(φA x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  {cab 2339  ∀wral 2614   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-int 3927 This theorem is referenced by:  ssmin  3945  ssintrab  3949  intmin4  3955
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