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Theorem rabid2 2667
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 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabid2
StepHypRef Expression
1 abeq2 2298 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
2 pm4.71 389 . . . 4 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴𝜑)))
32albii 1481 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
41, 3bitr4i 187 . 2 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴𝜑))
5 df-rab 2477 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65eqeq2i 2200 . 2 (𝐴 = {𝑥𝐴𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)})
7 df-ral 2473 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
84, 6, 73bitr4i 212 1 (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362   = wceq 1364  wcel 2160  {cab 2175  wral 2468  {crab 2472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-rab 2477
This theorem is referenced by:  rabxmdc  3469  rabrsndc  3675  class2seteq  4181  dmmptg  5144  dmmptd  5365  fneqeql  5644  fmpt  5686  acexmidlemph  5888  inffiexmid  6933  ssfirab  6961  exmidaclem  7236  ioomax  9977  iccmax  9978  dfphi2  12251  phiprmpw  12253  phisum  12271  unennn  12447  znnen  12448  isnsg4  13148  lssuni  13676
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