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Theorem iota4 5786
 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 2462 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 biimpr 209 . . . . . 6 ((𝜑𝑥 = 𝑧) → (𝑥 = 𝑧𝜑))
32alimi 1730 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧𝜑))
4 sb2 2340 . . . . 5 (∀𝑥(𝑥 = 𝑧𝜑) → [𝑧 / 𝑥]𝜑)
53, 4syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑)
6 iotaval 5779 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
76eqcomd 2616 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑))
8 dfsbcq2 3405 . . . . 5 (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
97, 8syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
105, 9mpbid 221 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
1110exlimiv 1845 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑)
121, 11sylbi 206 1 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695  [wsb 1867  ∃!weu 2458  [wsbc 3402  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768 This theorem is referenced by:  iota4an  5787  iotacl  5791  pm14.24  37655  sbiota1  37657
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