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Theorem zfrep3cl 5256
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 2904 . 2 𝑥𝐴
2 zfrep3cl.1 . 2 𝐴 ∈ V
3 zfrep3cl.2 . 2 (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))
41, 2, 3zfrepclf 5255 1 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  Vcvv 3447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886
This theorem is referenced by: (None)
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