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Theorem iota4 6309
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 2637 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 biimpr 223 . . . . . 6 ((𝜑𝑥 = 𝑧) → (𝑥 = 𝑧𝜑))
32alimi 1813 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧𝜑))
4 sb6 2091 . . . . 5 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
53, 4sylibr 237 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑)
6 iotaval 6302 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
76eqcomd 2807 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
8 dfsbcq2 3726 . . . . 5 (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
97, 8syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
105, 9mpbid 235 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
1110exlimiv 1931 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
121, 11sylbi 220 1 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wex 1781  [wsb 2069  ∃!weu 2631  [wsbc 3723  cio 6285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-uni 4804  df-iota 6287
This theorem is referenced by:  iota4an  6310  iotacl  6314  pm14.24  41123  sbiota1  41125  eubrdm  43615
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