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Theorem iotasbc2 42792
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 42791 . 2 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒)))
2 iotasbc 42791 . . . . 5 (∃!𝑥𝜓 → ([(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
32anbi2d 630 . . . 4 (∃!𝑥𝜓 → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒))))
4 3anass 1096 . . . . . 6 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
54exbii 1851 . . . . 5 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
6 19.42v 1958 . . . . 5 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
75, 6bitr2i 276 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒))
83, 7bitrdi 287 . . 3 (∃!𝑥𝜓 → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
98exbidv 1925 . 2 (∃!𝑥𝜓 → (∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑦𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
101, 9sylan9bb 511 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 397  w3a 1088  wal 1540  wex 1782  ∃!weu 2563  [wsbc 3743  cio 6450
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-sbc 3744  df-un 3919  df-in 3921  df-ss 3931  df-sn 4591  df-pr 4593  df-uni 4870  df-iota 6452
This theorem is referenced by: (None)
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