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Theorem ssrab 3012
 Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssrab (B ⊆ {x Aφ} ↔ (BA x B φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem ssrab
StepHypRef Expression
1 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
21sseq2i 2964 . 2 (B ⊆ {x Aφ} ↔ B ⊆ {x ∣ (x A φ)})
3 ssab 3004 . 2 (B ⊆ {x ∣ (x A φ)} ↔ x(x B → (x A φ)))
4 dfss3 2929 . . . 4 (BAx B x A)
54anbi1i 431 . . 3 ((BA x B φ) ↔ (x B x A x B φ))
6 r19.26 2435 . . 3 (x B (x A φ) ↔ (x B x A x B φ))
7 df-ral 2305 . . 3 (x B (x A φ) ↔ x(x B → (x A φ)))
85, 6, 73bitr2ri 198 . 2 (x(x B → (x A φ)) ↔ (BA x B φ))
92, 3, 83bitri 195 1 (B ⊆ {x Aφ} ↔ (BA x B φ))
 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 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-bndl 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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-in 2918  df-ss 2925 This theorem is referenced by:  ssrabdv  3013
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