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Theorem iota4 4828
Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iota4 (∃!xφ[(℩xφ) / x]φ)

Proof of Theorem iota4
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-eu 1900 . 2 (∃!xφzx(φx = z))
2 bi2 121 . . . . . 6 ((φx = z) → (x = zφ))
32alimi 1341 . . . . 5 (x(φx = z) → x(x = zφ))
4 sb2 1647 . . . . 5 (x(x = zφ) → [z / x]φ)
53, 4syl 14 . . . 4 (x(φx = z) → [z / x]φ)
6 iotaval 4821 . . . . . 6 (x(φx = z) → (℩xφ) = z)
76eqcomd 2042 . . . . 5 (x(φx = z) → z = (℩xφ))
8 dfsbcq2 2761 . . . . 5 (z = (℩xφ) → ([z / x]φ[(℩xφ) / x]φ))
97, 8syl 14 . . . 4 (x(φx = z) → ([z / x]φ[(℩xφ) / x]φ))
105, 9mpbid 135 . . 3 (x(φx = z) → [(℩xφ) / x]φ)
1110exlimiv 1486 . 2 (zx(φx = z) → [(℩xφ) / x]φ)
121, 11sylbi 114 1 (∃!xφ[(℩xφ) / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  wex 1378  [wsb 1642  ∃!weu 1897  [wsbc 2758  cio 4808
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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by:  iota4an  4829  iotacl  4833
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