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Theorem rabid2 2464
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 2128 . . 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 1339 . . 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 2293 . . 3 {x Aφ} = {x ∣ (x A φ)}
65eqeq2i 2032 . 2 (A = {x Aφ} ↔ A = {x ∣ (x A φ)})
7 df-ral 2289 . 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 1226   = wceq 1228   wcel 1374  {cab 2008  wral 2284  {crab 2288
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-rab 2293
This theorem is referenced by:  rabxmdc  3226  rabrsndc  3412  class2seteq  3890  dmmptg  4745  fneqeql  5200  fmpt  5244  acexmidlemph  5429
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