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Theorem ssintrab 3949
 Description: Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
Assertion
Ref Expression
ssintrab (A {x B φ} ↔ x B (φA x))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem ssintrab
StepHypRef Expression
1 df-rab 2623 . . . 4 {x B φ} = {x (x B φ)}
21inteqi 3930 . . 3 {x B φ} = {x (x B φ)}
32sseq2i 3296 . 2 (A {x B φ} ↔ A {x (x B φ)})
4 impexp 433 . . . 4 (((x B φ) → A x) ↔ (x B → (φA x)))
54albii 1566 . . 3 (x((x B φ) → A x) ↔ x(x B → (φA x)))
6 ssintab 3943 . . 3 (A {x (x B φ)} ↔ x((x B φ) → A x))
7 df-ral 2619 . . 3 (x B (φA x) ↔ x(x B → (φA x)))
85, 6, 73bitr4i 268 . 2 (A {x (x B φ)} ↔ x B (φA x))
93, 8bitri 240 1 (A {x B φ} ↔ x B (φA x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  {crab 2618   ⊆ 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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by: (None)
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