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Theorem zfrep3cl 5188
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 5187 1 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541  wex 1787  wcel 2110  Vcvv 3408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886
This theorem is referenced by: (None)
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