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Theorem rabid2f 3470
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
21eqabf 2937 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
3 pm4.71 557 . . . 4 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴𝜑)))
43albii 1817 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝜑)))
52, 4bitr4i 278 . 2 (𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐴𝜑))
6 df-rab 3439 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
76eqeq2i 2747 . 2 (𝐴 = {𝑥𝐴𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥𝐴𝜑)})
8 df-ral 3064 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
95, 7, 83bitr4i 303 1 (𝐴 = {𝑥𝐴𝜑} ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2103  {cab 2711  wnfc 2888  wral 3063  {crab 3438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rab 3439
This theorem is referenced by:  rabid2  3472  funcnvmpt  32677  dmmptdff  45064  dmmptdf2  45074
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