Theorem zfrep3cl 1478

Description: An inference rule based on the Axiom of Replacement. Typically, φ defines a function from x to y.

Hypotheses

Ref	Expression
zfrep3cl.1	⊢ A ∈ V
zfrep3cl.2	⊢ (x ∈ A → ∃z∀y(φ → y = z))

Assertion

Ref	Expression
zfrep3cl	⊢ ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ∧ φ))

Distinct variable group(s):   x,y,z,A   φ,z

Proof of Theorem zfrep3cl

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (y ∈ A → ∀x y ∈ A)
2		zfrep3cl.1	. 2 ⊢ A ∈ V
3		zfrep3cl.2	. 2 ⊢ (x ∈ A → ∃z∀y(φ → y = z))
4	1, 2, 3	zfrepclf 1477	1 ⊢ ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ∧ φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  zfaus 1480

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-12 802  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

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-v 1349

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