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Theorem nfsbt 1847
Description: Closed form of nfsb 1819. (Contributed by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsbt (xzφ → Ⅎz[y / x]φ)
Distinct variable group:   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem nfsbt
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1416 . 2 (xzφwxzφ)
2 nfsbxyt 1816 . . . . 5 (xzφ → Ⅎz[w / x]φ)
32alimi 1341 . . . 4 (wxzφwz[w / x]φ)
4 nfsbxyt 1816 . . . 4 (wz[w / x]φ → Ⅎz[y / w][w / x]φ)
53, 4syl 14 . . 3 (wxzφ → Ⅎz[y / w][w / x]φ)
6 nfv 1418 . . . . 5 wφ
76sbco2 1836 . . . 4 ([y / w][w / x]φ ↔ [y / x]φ)
87nfbii 1359 . . 3 (Ⅎz[y / w][w / x]φ ↔ Ⅎz[y / x]φ)
95, 8sylib 127 . 2 (wxzφ → Ⅎz[y / x]φ)
101, 9syl 14 1 (xzφ → Ⅎz[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  wnf 1346  [wsb 1642
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
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  nfsbd  1848  setindft  9395
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