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Theorem pm14.24 41588
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 2589 . . . . 5 𝑥∃!𝑥𝜑
2 nfsbc1v 3700 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
3 pm14.12 41577 . . . . . . . . . 10 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4319.21bbi 2191 . . . . . . . . 9 (∃!𝑥𝜑 → ((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
54ancomsd 469 . . . . . . . 8 (∃!𝑥𝜑 → (([𝑦 / 𝑥]𝜑𝜑) → 𝑥 = 𝑦))
65expdimp 456 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
7 pm13.13b 41564 . . . . . . . . 9 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) → 𝜑)
87ex 416 . . . . . . . 8 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
98adantl 485 . . . . . . 7 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝑥 = 𝑦𝜑))
106, 9impbid 215 . . . . . 6 ((∃!𝑥𝜑[𝑦 / 𝑥]𝜑) → (𝜑𝑥 = 𝑦))
1110ex 416 . . . . 5 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
121, 2, 11alrimd 2217 . . . 4 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
13 iotaval 6313 . . . . 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 6320 . . . 4 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
17 dfsbcq 3682 . . . 4 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
1816, 17syl5ibrcom 250 . . 3 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → [𝑦 / 𝑥]𝜑))
1915, 18impbid 215 . 2 (∃!𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
2019alrimiv 1934 1 (∃!𝑥𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = (℩𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540   = wceq 1542  ∃!weu 2569  [wsbc 3680  cio 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400  df-sbc 3681  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-uni 4797  df-iota 6297
This theorem is referenced by: (None)
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