Theorem rabid2f 3426

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	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
rabid2f	⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Proof of Theorem rabid2f

Step	Hyp	Ref	Expression
1		rabid2f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	eqabf 2924	. . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		pm4.71 557	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4	3	albii 1820	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
5	2, 4	bitr4i 278	. 2 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-rab 3396	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	6	eqeq2i 2744	. 2 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
8		df-ral 3048	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
9	5, 7, 8	3bitr4i 303	1 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111  {cab 2709  Ⅎwnfc 2879  ∀wral 3047  {crab 3395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rab 3396

This theorem is referenced by:  rabid2  3428  funcnvmpt  32649  dmmptdff  45330  dmmptdf2  45340