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Theorem iotasbc2 39147
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 39146 . 2 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒)))
2 iotasbc 39146 . . . . 5 (∃!𝑥𝜓 → ([(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
32anbi2d 614 . . . 4 (∃!𝑥𝜓 → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒))))
4 3anass 1080 . . . . . 6 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
54exbii 1924 . . . . 5 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
6 19.42v 2033 . . . . 5 (∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ (∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
75, 6bitr2i 265 . . . 4 ((∀𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒))
83, 7syl6bb 276 . . 3 (∃!𝑥𝜓 → ((∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
98exbidv 2002 . 2 (∃!𝑥𝜓 → (∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑦𝑧(∀𝑥(𝜑𝑥 = 𝑦) ∧ ∀𝑥(𝜓𝑥 = 𝑧) ∧ 𝜒)))
101, 9sylan9bb 499 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 196  wa 382  w3a 1071  wal 1629  wex 1852  ∃!weu 2618  [wsbc 3587  cio 5991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-sbc 3588  df-un 3728  df-sn 4318  df-pr 4320  df-uni 4576  df-iota 5993
This theorem is referenced by: (None)
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