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Theorem rabid2f 3307
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 2975 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
3 pm4.71 549 . . . 4 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴𝜑)))
43albii 1907 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
52, 4bitr4i 269 . 2 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴𝜑))
6 df-rab 3104 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
76eqeq2i 2817 . 2 (𝐴 = {𝑥𝐴𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)})
8 df-ral 3100 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
95, 7, 83bitr4i 294 1 (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wcel 2158  {cab 2791  wnfc 2934  wral 3095  {crab 3099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ral 3100  df-rab 3104
This theorem is referenced by:  funcnvmptOLD  29791  funcnvmpt  29792  dmmptdf2  39920
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