Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfrep3cl Structured version   Visualization version   GIF version

Theorem zfrep3cl 5167
 Description: An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
Hypotheses
Ref Expression
zfrep3cl.1 𝐴 ∈ V
zfrep3cl.2 (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))
Assertion
Ref Expression
zfrep3cl 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem zfrep3cl
StepHypRef Expression
1 nfcv 2955 . 2 𝑥𝐴
2 zfrep3cl.1 . 2 𝐴 ∈ V
3 zfrep3cl.2 . 2 (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))
41, 2, 3zfrepclf 5166 1 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  Vcvv 3442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870  df-nfc 2938 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator