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Theorem iota4 6329
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 eu6 2652 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 biimpr 221 . . . . . 6 ((𝜑𝑥 = 𝑧) → (𝑥 = 𝑧𝜑))
32alimi 1803 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧𝜑))
4 sb6 2084 . . . . 5 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
53, 4sylibr 235 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑)
6 iotaval 6322 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
76eqcomd 2824 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
8 dfsbcq2 3772 . . . . 5 (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
97, 8syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
105, 9mpbid 233 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
1110exlimiv 1922 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
121, 11sylbi 218 1 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526   = wceq 1528  wex 1771  [wsb 2060  ∃!weu 2646  [wsbc 3769  cio 6305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-sbc 3770  df-un 3938  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307
This theorem is referenced by:  iota4an  6330  iotacl  6334  pm14.24  40641  sbiota1  40643  eubrdm  43148
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