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Theorem axreg2 9537
Description: Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
axreg2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axreg2
StepHypRef Expression
1 ax-reg 9536 . 2 (∃𝑥 𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2119.23bi 2185 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540  wex 1782
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-12 2172  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-ex 1783
This theorem is referenced by:  zfregcl  9538  axregndlem2  10547
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