Theorem rabid2 1309

Description: An "identity" law for restricted class abstraction.

Assertion

Ref	Expression
rabid2	⊢ (A = {x ∈ A∣φ} ↔ ∀x ∈ A φ)

Distinct variable group(s):   x,A

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		pm4.71 481	. . . 4 ⊢ ((x ∈ A → φ) ↔ (x ∈ A ↔ (x ∈ A ∧ φ)))
2	1	bial 695	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ)))
3		cleqab 1174	. . 3 ⊢ (A = {x∣(x ∈ A ∧ φ)} ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ φ)))
4	2, 3	bitr4 154	. 2 ⊢ (∀x(x ∈ A → φ) ↔ A = {x∣(x ∈ A ∧ φ)})
5		df-ral 1205	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		df-rab 1208	. . 3 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ∧ φ)}
7	6	cleq2i 1111	. 2 ⊢ (A = {x ∈ A∣φ} ↔ A = {x∣(x ∈ A ∧ φ)})
8	4, 5, 7	3bitr4r 159	1 ⊢ (A = {x ∈ A∣φ} ↔ ∀x ∈ A φ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204

This theorem is referenced by:  zfrep6 2744  abrexex 2912

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rab 1208

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