Theorem rabid2 2710

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 2340	. . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2		pm4.71 389	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3	2	albii 1518	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4	1, 3	bitr4i 187	. 2 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5		df-rab 2519	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	eqeq2i 2242	. 2 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
7		df-ral 2515	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	4, 6, 7	3bitr4i 212	1 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510  {crab 2514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-rab 2519

This theorem is referenced by:  rabxmdc  3526  rabrsndc  3739  class2seteq  4253  dmmptg  5234  dmmptd  5463  fneqeql  5755  fmpt  5797  acexmidlemph  6010  inffiexmid  7097  ssfirab  7128  exmidaclem  7422  ioomax  10182  iccmax  10183  dfphi2  12791  phiprmpw  12793  phisum  12812  unennn  13017  znnen  13018  isnsg4  13798  lssuni  14376  psrbagfi  14686  lgsquadlem2  15806