Theorem rabid2 2464

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 2128	. . 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 1339	. . 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 2293	. . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
6	5	eqeq2i 2032	. 2 ⊢ (A = {x ∈ A ∣ φ} ↔ A = {x ∣ (x ∈ A ∧ φ)})
7		df-ral 2289	. 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 1226   = wceq 1228   ∈ wcel 1374  {cab 2008  ∀wral 2284  {crab 2288

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-rab 2293

This theorem is referenced by:  rabxmdc  3226  rabrsndc  3412  class2seteq  3890  dmmptg  4745  fneqeql  5200  fmpt  5244  acexmidlemph  5429