Theorem clabel 1579

Description: Membership of a class abstraction in another class

Assertion

Ref	Expression
clabel	⊢ ({x∣φ} ∈ A ↔ ∃y(y ∈ A ⋀ ∀x(x ∈ y ↔ φ)))

Distinct variable groups:   y,A   φ,y   x,y

Proof of Theorem clabel

Step	Hyp	Ref	Expression
1		df-clel 1470	. 2 ⊢ ({x∣φ} ∈ A ↔ ∃y(y = {x∣φ} ⋀ y ∈ A))
2		abeq2 1565	. . . . 5 ⊢ (y = {x∣φ} ↔ ∀x(x ∈ y ↔ φ))
3	2	anbi1i 481	. . . 4 ⊢ ((y = {x∣φ} ⋀ y ∈ A) ↔ (∀x(x ∈ y ↔ φ) ⋀ y ∈ A))
4		ancom 435	. . . 4 ⊢ ((∀x(x ∈ y ↔ φ) ⋀ y ∈ A) ↔ (y ∈ A ⋀ ∀x(x ∈ y ↔ φ)))
5	3, 4	bitr 173	. . 3 ⊢ ((y = {x∣φ} ⋀ y ∈ A) ↔ (y ∈ A ⋀ ∀x(x ∈ y ↔ φ)))
6	5	exbii 1049	. 2 ⊢ (∃y(y = {x∣φ} ⋀ y ∈ A) ↔ ∃y(y ∈ A ⋀ ∀x(x ∈ y ↔ φ)))
7	1, 6	bitr 173	1 ⊢ ({x∣φ} ∈ A ↔ ∃y(y ∈ A ⋀ ∀x(x ∈ y ↔ φ)))

Syntax hints:   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  {cab 1461

This theorem is referenced by:  grothprimlem 8721

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

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