ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  repizf Structured version   GIF version

Theorem repizf 3864
Description: Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 3863. It is identical to zfrep6 3865 except for the choice of a freeness hypothesis rather than a distinct variable constraint between 𝑏 and φ. (Contributed by Jim Kingdon, 23-Aug-2018.)
Hypothesis
Ref Expression
ax-coll.1 𝑏φ
Assertion
Ref Expression
repizf (x 𝑎 ∃!yφ𝑏x 𝑎 y 𝑏 φ)
Distinct variable group:   x,y,𝑎,𝑏
Allowed substitution hints:   φ(x,y,𝑎,𝑏)

Proof of Theorem repizf
StepHypRef Expression
1 euex 1927 . . 3 (∃!yφyφ)
21ralimi 2378 . 2 (x 𝑎 ∃!yφx 𝑎 yφ)
3 ax-coll.1 . . 3 𝑏φ
43ax-coll 3863 . 2 (x 𝑎 yφ𝑏x 𝑎 y 𝑏 φ)
52, 4syl 14 1 (x 𝑎 ∃!yφ𝑏x 𝑎 y 𝑏 φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1346  wex 1378  ∃!weu 1897  wral 2300  wrex 2301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-coll 3863
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-ral 2305
This theorem is referenced by:  zfrep6  3865  repizf2  3906
  Copyright terms: Public domain W3C validator