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Theorem pm14.24 38453
Description: Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
pm14.24 (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm14.24
StepHypRef Expression
1 nfeu1 2478 . . . . 5 𝑥∃!𝑥𝜑
2 nfsbc1v 3449 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
3 pm14.12 38442 . . . . . . . . . 10 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4319.21bbi 2058 . . . . . . . . 9 (∃!𝑥𝜑 → ((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
54ancomsd 470 . . . . . . . 8 (∃!𝑥𝜑 → (([𝑦 / 𝑥]𝜑𝜑) → 𝑥 = 𝑦))
65expdimp 453 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
7 pm13.13b 38429 . . . . . . . . 9 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) → 𝜑)
87ex 450 . . . . . . . 8 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
98adantl 482 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝑥 = 𝑦𝜑))
106, 9impbid 202 . . . . . 6 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
1110ex 450 . . . . 5 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
121, 2, 11alrimd 2082 . . . 4 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
13 iotaval 5850 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
1413eqcomd 2626 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
1512, 14syl6 35 . . 3 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
16 iota4 5857 . . . 4 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
17 dfsbcq 3431 . . . 4 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
1816, 17syl5ibrcom 237 . . 3 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → [𝑦 / 𝑥]𝜑))
1915, 18impbid 202 . 2 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
2019alrimiv 1853 1 (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  ∃!weu 2468  [wsbc 3429  cio 5837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-v 3197  df-sbc 3430  df-un 3572  df-sn 4169  df-pr 4171  df-uni 4428  df-iota 5839
This theorem is referenced by: (None)
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