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Theorem pm14.24 43493
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 2580 . . . . 5 𝑥∃!𝑥𝜑
2 nfsbc1v 3796 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
3 pm14.12 43482 . . . . . . . . . 10 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4319.21bbi 2181 . . . . . . . . 9 (∃!𝑥𝜑 → ((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
54ancomsd 464 . . . . . . . 8 (∃!𝑥𝜑 → (([𝑦 / 𝑥]𝜑𝜑) → 𝑥 = 𝑦))
65expdimp 451 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
7 pm13.13b 43469 . . . . . . . . 9 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) → 𝜑)
87ex 411 . . . . . . . 8 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
98adantl 480 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝑥 = 𝑦𝜑))
106, 9impbid 211 . . . . . 6 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
1110ex 411 . . . . 5 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
121, 2, 11alrimd 2206 . . . 4 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
13 iotaval 6513 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
1413eqcomd 2736 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
1512, 14syl6 35 . . 3 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
16 iota4 6523 . . . 4 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
17 dfsbcq 3778 . . . 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 1928 1 (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537   = wceq 1539  ∃!weu 2560  [wsbc 3776  cio 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-v 3474  df-sbc 3777  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6494
This theorem is referenced by: (None)
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