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Axiom ax-coll 3899
Description: Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3952 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.)
Hypothesis
Ref Expression
ax-coll.1  |-  F/ b
ph
Assertion
Ref Expression
ax-coll  |-  ( A. x  e.  a  E. y ph  ->  E. b A. x  e.  a  E. y  e.  b  ph )
Distinct variable group:    x, y, a, b
Allowed substitution hints:    ph( x, y, a, b)

Detailed syntax breakdown of Axiom ax-coll
StepHypRef Expression
1 wph . . . 4  wff  ph
2 vy . . . 4  setvar  y
31, 2wex 1397 . . 3  wff  E. y ph
4 vx . . 3  setvar  x
5 va . . . 4  setvar  a
65cv 1258 . . 3  class  a
73, 4, 6wral 2323 . 2  wff  A. x  e.  a  E. y ph
8 vb . . . . . 6  setvar  b
98cv 1258 . . . . 5  class  b
101, 2, 9wrex 2324 . . . 4  wff  E. y  e.  b  ph
1110, 4, 6wral 2323 . . 3  wff  A. x  e.  a  E. y  e.  b  ph
1211, 8wex 1397 . 2  wff  E. b A. x  e.  a  E. y  e.  b  ph
137, 12wi 4 1  wff  ( A. x  e.  a  E. y ph  ->  E. b A. x  e.  a  E. y  e.  b  ph )
Colors of variables: wff set class
This axiom is referenced by:  repizf  3900  bnd  3952
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