Theorem rabid2 1773

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

Assertion

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

Distinct variable group:   x,A

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		pm4.71 637	. . . 4 ⊢ ((x ∈ A → φ) ↔ (x ∈ A ↔ (x ∈ A ⋀ φ)))
2	1	albii 1001	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ A ↔ (x ∈ A ⋀ φ)))
3		abeq2 1571	. . 3 ⊢ (A = {x∣(x ∈ A ⋀ φ)} ↔ ∀x(x ∈ A ↔ (x ∈ A ⋀ φ)))
4	2, 3	bitr4 176	. 2 ⊢ (∀x(x ∈ A → φ) ↔ A = {x∣(x ∈ A ⋀ φ)})
5		df-ral 1652	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		df-rab 1655	. . 3 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ⋀ φ)}
7	6	eqeq2i 1488	. 2 ⊢ (A = {x ∈ A∣φ} ↔ A = {x∣(x ∈ A ⋀ φ)})
8	4, 5, 7	3bitr4r 184	1 ⊢ (A = {x ∈ A∣φ} ↔ ∀x ∈ A φ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  {cab 1466  ∀wral 1648  {crab 1651

This theorem is referenced by:  class2seteq 2740  zfrep6 3620  abrexex 3866  ioomax 6394

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rab 1655

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