Theorem rabid2 2480

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 = {x ∈ A ∣ φ} ↔ ∀x ∈ A φ)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		abeq2 2143	. . 3 ⊢ (A = {x ∣ (x ∈ A ∧ φ)} ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ)))
2		pm4.71 369	. . . 4 ⊢ ((x ∈ A → φ) ↔ (x ∈ A ↔ (x ∈ A ∧ φ)))
3	2	albii 1356	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ)))
4	1, 3	bitr4i 176	. 2 ⊢ (A = {x ∣ (x ∈ A ∧ φ)} ↔ ∀x(x ∈ A → φ))
5		df-rab 2309	. . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
6	5	eqeq2i 2047	. 2 ⊢ (A = {x ∈ A ∣ φ} ↔ A = {x ∣ (x ∈ A ∧ φ)})
7		df-ral 2305	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
8	4, 6, 7	3bitr4i 201	1 ⊢ (A = {x ∈ A ∣ φ} ↔ ∀x ∈ A φ)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  {crab 2304

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rab 2309

This theorem is referenced by:  rabxmdc  3243  rabrsndc  3429  class2seteq  3907  dmmptg  4761  fneqeql  5218  fmpt  5262  acexmidlemph  5448  ioomax  8587  iccmax  8588