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Theorem ssintrab 3629
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 2309 . . . 4 {x Bφ} = {x ∣ (x B φ)}
21inteqi 3610 . . 3 {x Bφ} = {x ∣ (x B φ)}
32sseq2i 2964 . 2 (A {x Bφ} ↔ A {x ∣ (x B φ)})
4 impexp 250 . . . 4 (((x B φ) → Ax) ↔ (x B → (φAx)))
54albii 1356 . . 3 (x((x B φ) → Ax) ↔ x(x B → (φAx)))
6 ssintab 3623 . . 3 (A {x ∣ (x B φ)} ↔ x((x B φ) → Ax))
7 df-ral 2305 . . 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 1240   wcel 1390  {cab 2023  wral 2300  {crab 2304  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-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-int 3607
This theorem is referenced by: (None)
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