Theorem zfauscl 2700

Description: Separation Scheme using a class variable. To derive this from ax-sep 2698, we invoke the Axiom of Extensionality (indirectly via vtocl 1838), which is needed for the justification of class variable notation.

If we omit the requirement that y not occur in φ, we can derive a contradiction, as notzfaus 2736 shows.

Hypothesis

Ref	Expression
zfauscl.1	⊢ A ∈ V

Assertion

Ref	Expression
zfauscl	⊢ ∃y∀x(x ∈ y ↔ (x ∈ A ⋀ φ))

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

Proof of Theorem zfauscl

Step	Hyp	Ref	Expression
1		zfauscl.1	. 2 ⊢ A ∈ V
2		eleq2 1532	. . . . . 6 ⊢ (z = A → (x ∈ z ↔ x ∈ A))
3	2	anbi1d 616	. . . . 5 ⊢ (z = A → ((x ∈ z ⋀ φ) ↔ (x ∈ A ⋀ φ)))
4	3	bibi2d 617	. . . 4 ⊢ (z = A → ((x ∈ y ↔ (x ∈ z ⋀ φ)) ↔ (x ∈ y ↔ (x ∈ A ⋀ φ))))
5	4	albidv 1276	. . 3 ⊢ (z = A → (∀x(x ∈ y ↔ (x ∈ z ⋀ φ)) ↔ ∀x(x ∈ y ↔ (x ∈ A ⋀ φ))))
6	5	exbidv 1277	. 2 ⊢ (z = A → (∃y∀x(x ∈ y ↔ (x ∈ z ⋀ φ)) ↔ ∃y∀x(x ∈ y ↔ (x ∈ A ⋀ φ))))
7		ax-sep 2698	. 2 ⊢ ∃y∀x(x ∈ y ↔ (x ∈ z ⋀ φ))
8	1, 6, 7	vtocl 1838	1 ⊢ ∃y∀x(x ∈ y ↔ (x ∈ A ⋀ φ))

Syntax hints:   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  Vcvv 1807

This theorem is referenced by:  nalset 2707  inex1 2711

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457  ax-sep 2698

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-v 1808

Copyright terms: Public domain