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Theorem iotasbc2 42680
Description: Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotasbc2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧   𝜓,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem iotasbc2
StepHypRef Expression
1 iotasbc 42679 . 2 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒)))
2 iotasbc 42679 . . . . 5 (∃!𝑥𝜓 → ([(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
32anbi2d 629 . . . 4 (∃!𝑥𝜓 → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒))))
4 3anass 1095 . . . . . 6 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
54exbii 1850 . . . . 5 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
6 19.42v 1957 . . . . 5 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
75, 6bitr2i 275 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒))
83, 7bitrdi 286 . . 3 (∃!𝑥𝜓 → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
98exbidv 1924 . 2 (∃!𝑥𝜓 → (∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑦𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
101, 9sylan9bb 510 1 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539  wex 1781  ∃!weu 2566  [wsbc 3738  cio 6444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3446  df-sbc 3739  df-un 3914  df-in 3916  df-ss 3926  df-sn 4586  df-pr 4588  df-uni 4865  df-iota 6446
This theorem is referenced by: (None)
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