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Theorem nfsbxyt 1816
Description: Closed form of nfsbxy 1815. (Contributed by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsbxyt (xzφ → Ⅎz[y / x]φ)
Distinct variable groups:   x,y   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem nfsbxyt
StepHypRef Expression
1 ax-bnd 1396 . 2 (z z = x (z z = y xz(x = yz x = y)))
2 nfs1v 1812 . . . . 5 z[y / z]φ
3 drsb1 1677 . . . . . 6 (z z = x → ([y / z]φ ↔ [y / x]φ))
43drnf2 1619 . . . . 5 (z z = x → (Ⅎz[y / z]φ ↔ Ⅎz[y / x]φ))
52, 4mpbii 136 . . . 4 (z z = x → Ⅎz[y / x]φ)
65a1d 22 . . 3 (z z = x → (xzφ → Ⅎz[y / x]φ))
7 a16nf 1743 . . . . 5 (z z = y → Ⅎz[y / x]φ)
87a1d 22 . . . 4 (z z = y → (xzφ → Ⅎz[y / x]φ))
9 df-nf 1347 . . . . . 6 (Ⅎz x = yz(x = yz x = y))
109albii 1356 . . . . 5 (xz x = yxz(x = yz x = y))
11 sb5 1764 . . . . . . 7 ([y / x]φx(x = y φ))
12 nfa1 1431 . . . . . . . . 9 xxz x = y
13 nfa1 1431 . . . . . . . . 9 xxzφ
1412, 13nfan 1454 . . . . . . . 8 x(xz x = y xzφ)
15 sp 1398 . . . . . . . . . 10 (xz x = y → Ⅎz x = y)
1615adantr 261 . . . . . . . . 9 ((xz x = y xzφ) → Ⅎz x = y)
17 sp 1398 . . . . . . . . . 10 (xzφ → Ⅎzφ)
1817adantl 262 . . . . . . . . 9 ((xz x = y xzφ) → Ⅎzφ)
1916, 18nfand 1457 . . . . . . . 8 ((xz x = y xzφ) → Ⅎz(x = y φ))
2014, 19nfexd 1641 . . . . . . 7 ((xz x = y xzφ) → Ⅎzx(x = y φ))
2111, 20nfxfrd 1361 . . . . . 6 ((xz x = y xzφ) → Ⅎz[y / x]φ)
2221ex 108 . . . . 5 (xz x = y → (xzφ → Ⅎz[y / x]φ))
2310, 22sylbir 125 . . . 4 (xz(x = yz x = y) → (xzφ → Ⅎz[y / x]φ))
248, 23jaoi 635 . . 3 ((z z = y xz(x = yz x = y)) → (xzφ → Ⅎz[y / x]φ))
256, 24jaoi 635 . 2 ((z z = x (z z = y xz(x = yz x = y))) → (xzφ → Ⅎz[y / x]φ))
261, 25ax-mp 7 1 (xzφ → Ⅎz[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628  wal 1240  wnf 1346  wex 1378  [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
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  nfsbt  1847
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