New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iota4 GIF version

Theorem iota4 4357
 Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iota4 (∃!xφ → [̣(℩xφ) / xφ)

Proof of Theorem iota4
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2208 . 2 (∃!xφzx(φx = z))
2 bi2 189 . . . . . 6 ((φx = z) → (x = zφ))
32alimi 1559 . . . . 5 (x(φx = z) → x(x = zφ))
4 sb2 2023 . . . . 5 (x(x = zφ) → [z / x]φ)
53, 4syl 15 . . . 4 (x(φx = z) → [z / x]φ)
6 iotaval 4350 . . . . . 6 (x(φx = z) → (℩xφ) = z)
76eqcomd 2358 . . . . 5 (x(φx = z) → z = (℩xφ))
8 dfsbcq2 3049 . . . . 5 (z = (℩xφ) → ([z / x]φ ↔ [̣(℩xφ) / xφ))
97, 8syl 15 . . . 4 (x(φx = z) → ([z / x]φ ↔ [̣(℩xφ) / xφ))
105, 9mpbid 201 . . 3 (x(φx = z) → [̣(℩xφ) / xφ)
1110exlimiv 1634 . 2 (zx(φx = z) → [̣(℩xφ) / xφ)
121, 11sylbi 187 1 (∃!xφ → [̣(℩xφ) / xφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃!weu 2204  [̣wsbc 3046  ℩cio 4337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  iota4an  4358  iotacl  4362
 Copyright terms: Public domain W3C validator