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Theorem rabid2f 3383
Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
Hypothesis
Ref Expression
rabid2f.1 𝑥𝐴
Assertion
Ref Expression
rabid2f (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)

Proof of Theorem rabid2f
StepHypRef Expression
1 rabid2f.1 . . . 4 𝑥𝐴
21abeq2f 3013 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
3 pm4.71 558 . . . 4 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴𝜑)))
43albii 1811 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
52, 4bitr4i 279 . 2 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴𝜑))
6 df-rab 3147 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
76eqeq2i 2834 . 2 (𝐴 = {𝑥𝐴𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)})
8 df-ral 3143 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
95, 7, 83bitr4i 304 1 (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  {cab 2799  wnfc 2961  wral 3138  {crab 3142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147
This theorem is referenced by:  funcnvmpt  30341  dmmptdf2  41383
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