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

Theorem nfsbxy 1800
Description: Similar to hbsb 1805 but with an extra distinct variable constraint, on x and y. (Contributed by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
nfsbxy.1 zφ
Assertion
Ref Expression
nfsbxy z[y / x]φ
Distinct variable groups:   x,y   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem nfsbxy
StepHypRef Expression
1 ax-bnd 1380 . 2 (z z = x (z z = y xz(x = yz x = y)))
2 nfs1v 1797 . . . 4 z[y / z]φ
3 drsb1 1662 . . . . 5 (z z = x → ([y / z]φ ↔ [y / x]φ))
43drnf2 1604 . . . 4 (z z = x → (Ⅎz[y / z]φ ↔ Ⅎz[y / x]φ))
52, 4mpbii 136 . . 3 (z z = x → Ⅎz[y / x]φ)
6 a16nf 1728 . . . 4 (z z = y → Ⅎz[y / x]φ)
7 df-nf 1330 . . . . . 6 (Ⅎz x = yz(x = yz x = y))
87albii 1339 . . . . 5 (xz x = yxz(x = yz x = y))
9 sb5 1749 . . . . . 6 ([y / x]φx(x = y φ))
10 nfa1 1416 . . . . . . 7 xxz x = y
11 sp 1382 . . . . . . . 8 (xz x = y → Ⅎz x = y)
12 nfsbxy.1 . . . . . . . . 9 zφ
1312a1i 9 . . . . . . . 8 (xz x = y → Ⅎzφ)
1411, 13nfand 1442 . . . . . . 7 (xz x = y → Ⅎz(x = y φ))
1510, 14nfexd 1626 . . . . . 6 (xz x = y → Ⅎzx(x = y φ))
169, 15nfxfrd 1344 . . . . 5 (xz x = y → Ⅎz[y / x]φ)
178, 16sylbir 125 . . . 4 (xz(x = yz x = y) → Ⅎz[y / x]φ)
186, 17jaoi 623 . . 3 ((z z = y xz(x = yz x = y)) → Ⅎz[y / x]φ)
195, 18jaoi 623 . 2 ((z z = x (z z = y xz(x = yz x = y))) → Ⅎz[y / x]φ)
201, 19ax-mp 7 1 z[y / x]φ
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616  wal 1226  wnf 1329  wex 1362  [wsb 1627
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628
This theorem is referenced by:  nfsb  1804  sbalyz  1857  opelopabsb  3971
  Copyright terms: Public domain W3C validator