Theorem rabid2 1767

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		pm4.71 634	. . . 4 ⊢ ((x ∈ A → φ) ↔ (x ∈ A ↔ (x ∈ A ⋀ φ)))
2	1	albii 997	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ A ↔ (x ∈ A ⋀ φ)))
3		abeq2 1565	. . 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 1646	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
6		df-rab 1649	. . 3 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ⋀ φ)}
7	6	eqeq2i 1482	. 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 952   = wceq 954   ∈ wcel 956  {cab 1461  ∀wral 1642  {crab 1645

This theorem is referenced by:  class2seteq 2730  zfrep6 3606  abrexex 3851  ioomax 6333

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rab 1649

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