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Theorem bnd 4091
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 4040). Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 4038. (Contributed by NM, 17-Oct-2004.)
Ref Expression
bnd (∀𝑥𝑧𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
Distinct variable groups:   𝜑,𝑧,𝑤   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bnd
StepHypRef Expression
1 nfv 1508 . 2 𝑤𝜑
21ax-coll 4038 1 (∀𝑥𝑧𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1468  wral 2414  wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425  ax-17 1506  ax-coll 4038
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  bnd2  4092
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