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Theorem pm14.24 42050
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 2588 . . . . 5 𝑥∃!𝑥𝜑
2 nfsbc1v 3736 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
3 pm14.12 42039 . . . . . . . . . 10 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4319.21bbi 2183 . . . . . . . . 9 (∃!𝑥𝜑 → ((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
54ancomsd 466 . . . . . . . 8 (∃!𝑥𝜑 → (([𝑦 / 𝑥]𝜑𝜑) → 𝑥 = 𝑦))
65expdimp 453 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
7 pm13.13b 42026 . . . . . . . . 9 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) → 𝜑)
87ex 413 . . . . . . . 8 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
98adantl 482 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝑥 = 𝑦𝜑))
106, 9impbid 211 . . . . . 6 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
1110ex 413 . . . . 5 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
121, 2, 11alrimd 2208 . . . 4 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
13 iotaval 6407 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
1413eqcomd 2744 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
1512, 14syl6 35 . . 3 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
16 iota4 6414 . . . 4 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
17 dfsbcq 3718 . . . 4 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
1816, 17syl5ibrcom 246 . . 3 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → [𝑦 / 𝑥]𝜑))
1915, 18impbid 211 . 2 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
2019alrimiv 1930 1 (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1537   = wceq 1539  ∃!weu 2568  [wsbc 3716  cio 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434  df-sbc 3717  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391
This theorem is referenced by: (None)
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