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Theorem iota4 6544
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 2572 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 biimpr 220 . . . . . 6 ((𝜑𝑥 = 𝑧) → (𝑥 = 𝑧𝜑))
32alimi 1808 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧𝜑))
4 sb6 2083 . . . . 5 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
53, 4sylibr 234 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑)
6 iotaval 6534 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
76eqcomd 2741 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
8 dfsbcq2 3794 . . . . 5 (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
97, 8syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
105, 9mpbid 232 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
1110exlimiv 1928 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
121, 11sylbi 217 1 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  [wsb 2062  ∃!weu 2566  [wsbc 3791  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sbc 3792  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by:  iota4an  6545  iotacl  6549  pm14.24  44428  sbiota1  44430  eubrdm  46986
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