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Theorem iota4 6012
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 2621 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 biimpr 210 . . . . . 6 ((𝜑𝑥 = 𝑧) → (𝑥 = 𝑧𝜑))
32alimi 1886 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧𝜑))
4 sb2 2497 . . . . 5 (∀𝑥(𝑥 = 𝑧𝜑) → [𝑧 / 𝑥]𝜑)
53, 4syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑)
6 iotaval 6005 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
76eqcomd 2776 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
8 dfsbcq2 3588 . . . . 5 (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
97, 8syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
105, 9mpbid 222 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
1110exlimiv 2009 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
121, 11sylbi 207 1 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628   = wceq 1630  wex 1851  [wsb 2048  ∃!weu 2617  [wsbc 3585  cio 5992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-v 3351  df-sbc 3586  df-un 3726  df-sn 4315  df-pr 4317  df-uni 4573  df-iota 5994
This theorem is referenced by:  iota4an  6013  iotacl  6017  pm14.24  39152  sbiota1  39154
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