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Theorem ssintrab 3608
 Description: Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
Assertion
Ref Expression
ssintrab (A {x Bφ} ↔ x B (φAx))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem ssintrab
StepHypRef Expression
1 df-rab 2289 . . . 4 {x Bφ} = {x ∣ (x B φ)}
21inteqi 3589 . . 3 {x Bφ} = {x ∣ (x B φ)}
32sseq2i 2943 . 2 (A {x Bφ} ↔ A {x ∣ (x B φ)})
4 impexp 250 . . . 4 (((x B φ) → Ax) ↔ (x B → (φAx)))
54albii 1335 . . 3 (x((x B φ) → Ax) ↔ x(x B → (φAx)))
6 ssintab 3602 . . 3 (A {x ∣ (x B φ)} ↔ x((x B φ) → Ax))
7 df-ral 2285 . . 3 (x B (φAx) ↔ x(x B → (φAx)))
85, 6, 73bitr4i 201 . 2 (A {x ∣ (x B φ)} ↔ x B (φAx))
93, 8bitri 173 1 (A {x Bφ} ↔ x B (φAx))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1224   ∈ wcel 1370  {cab 2004  ∀wral 2280  {crab 2284   ⊆ wss 2890  ∩ cint 3585 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rab 2289  df-v 2533  df-in 2897  df-ss 2904  df-int 3586 This theorem is referenced by: (None)
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