Theorem zfrep6 4166

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 4167 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 1552	. 2 ⊢ Ⅎ𝑤𝜑
2	1	repizf 4165	1 ⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4  ∃wex 1516  ∃!weu 2055  ∀wral 2485  ∃wrex 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-coll 4164

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-ral 2490

This theorem is referenced by:  funimaexglem  5363