Theorem zfrep6 4045

Description: A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4046 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version. (Contributed by NM, 10-Oct-2003.)

Assertion

Ref	Expression
zfrep6	⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Distinct variable groups:   𝜑,𝑤   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		nfv 1508	. 2 ⊢ Ⅎ𝑤𝜑
2	1	repizf 4044	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4  ∃wex 1468  ∃!weu 1999  ∀wral 2416  ∃wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-coll 4043

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-ral 2421

This theorem is referenced by:  funimaexglem  5206