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Theorem rabid2 2480
Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rabid2 (A = {x Aφ} ↔ x A φ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rabid2
StepHypRef Expression
1 abeq2 2143 . . 3 (A = {x ∣ (x A φ)} ↔ x(x A ↔ (x A φ)))
2 pm4.71 369 . . . 4 ((x Aφ) ↔ (x A ↔ (x A φ)))
32albii 1356 . . 3 (x(x Aφ) ↔ x(x A ↔ (x A φ)))
41, 3bitr4i 176 . 2 (A = {x ∣ (x A φ)} ↔ x(x Aφ))
5 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
65eqeq2i 2047 . 2 (A = {x Aφ} ↔ A = {x ∣ (x A φ)})
7 df-ral 2305 . 2 (x A φx(x Aφ))
84, 6, 73bitr4i 201 1 (A = {x Aφ} ↔ x A φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wral 2300  {crab 2304
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-ral 2305  df-rab 2309
This theorem is referenced by:  rabxmdc  3243  rabrsndc  3429  class2seteq  3907  dmmptg  4761  fneqeql  5218  fmpt  5262  acexmidlemph  5448  ioomax  8547  iccmax  8548
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