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Theorem zfrep3cl 4938
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 2907 . 2 𝑥𝐴
2 zfrep3cl.1 . 2 𝐴 ∈ V
3 zfrep3cl.2 . 2 (𝑥𝐴 → ∃𝑧𝑦(𝜑𝑦 = 𝑧))
41, 2, 3zfrepclf 4937 1 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  wcel 2155  Vcvv 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352
This theorem is referenced by: (None)
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