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Theorem rabid2 2503
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 2162 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
2 pm4.71 375 . . . 4 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴𝜑)))
32albii 1375 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
41, 3bitr4i 180 . 2 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴𝜑))
5 df-rab 2332 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65eqeq2i 2066 . 2 (𝐴 = {𝑥𝐴𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)})
7 df-ral 2328 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
84, 6, 73bitr4i 205 1 (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wral 2323  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-rab 2332
This theorem is referenced by:  rabxmdc  3276  rabrsndc  3465  class2seteq  3943  dmmptg  4845  fneqeql  5302  fmpt  5346  acexmidlemph  5532  ioomax  8917  iccmax  8918
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