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Theorem nfsbxyt 1801
 Description: Closed form of nfsbxy 1800. (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 1380 . 2 (z z = x (z z = y xz(x = yz x = y)))
2 nfs1v 1797 . . . . 5 z[y / z]φ
3 drsb1 1662 . . . . . 6 (z z = x → ([y / z]φ ↔ [y / x]φ))
43drnf2 1604 . . . . 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 1728 . . . . 5 (z z = y → Ⅎz[y / x]φ)
87a1d 22 . . . 4 (z z = y → (xzφ → Ⅎz[y / x]φ))
9 df-nf 1330 . . . . . 6 (Ⅎz x = yz(x = yz x = y))
109albii 1339 . . . . 5 (xz x = yxz(x = yz x = y))
11 sb5 1749 . . . . . . 7 ([y / x]φx(x = y φ))
12 nfa1 1416 . . . . . . . . 9 xxz x = y
13 nfa1 1416 . . . . . . . . 9 xxzφ
1412, 13nfan 1439 . . . . . . . 8 x(xz x = y xzφ)
15 sp 1382 . . . . . . . . . 10 (xz x = y → Ⅎz x = y)
1615adantr 261 . . . . . . . . 9 ((xz x = y xzφ) → Ⅎz x = y)
17 sp 1382 . . . . . . . . . 10 (xzφ → Ⅎzφ)
1817adantl 262 . . . . . . . . 9 ((xz x = y xzφ) → Ⅎzφ)
1916, 18nfand 1442 . . . . . . . 8 ((xz x = y xzφ) → Ⅎz(x = y φ))
2014, 19nfexd 1626 . . . . . . 7 ((xz x = y xzφ) → Ⅎzx(x = y φ))
2111, 20nfxfrd 1344 . . . . . 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 623 . . 3 ((z z = y xz(x = yz x = y)) → (xzφ → Ⅎz[y / x]φ))
256, 24jaoi 623 . 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 616  ∀wal 1226  Ⅎwnf 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:  nfsbt  1832
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