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Theorem vtoclgft 3442
Description: Closed theorem form of vtoclgf 3451. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.)
Assertion
Ref Expression
vtoclgft (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)

Proof of Theorem vtoclgft
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elisset 3403 . . . . 5 (𝐴𝑉 → ∃𝑧 𝑧 = 𝐴)
2 nfnfc1 2944 . . . . . 6 𝑥𝑥𝐴
3 nfcvd 2942 . . . . . . 7 (𝑥𝐴𝑥𝑧)
4 id 22 . . . . . . 7 (𝑥𝐴𝑥𝐴)
53, 4nfeqd 2949 . . . . . 6 (𝑥𝐴 → Ⅎ𝑥 𝑧 = 𝐴)
6 eqeq1 2803 . . . . . . 7 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
76a1i 11 . . . . . 6 (𝑥𝐴 → (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴)))
82, 5, 7cbvexd 2415 . . . . 5 (𝑥𝐴 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
91, 8syl5ib 236 . . . 4 (𝑥𝐴 → (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴))
109ad2antrr 718 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴))
11103impia 1146 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → ∃𝑥 𝑥 = 𝐴)
12 biimp 207 . . . . . . . . 9 ((𝜑𝜓) → (𝜑𝜓))
1312imim2i 16 . . . . . . . 8 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
1413com23 86 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜑 → (𝑥 = 𝐴𝜓)))
1514imp 396 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
1615alanimi 1912 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴𝜓))
17 19.23t 2244 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
1817adantl 474 . . . . 5 ((𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
1916, 18syl5ib 236 . . . 4 ((𝑥𝐴 ∧ Ⅎ𝑥𝜓) → ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝐴𝜓)))
2019imp 396 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑥 𝑥 = 𝐴𝜓))
21203adant3 1163 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → (∃𝑥 𝑥 = 𝐴𝜓))
2211, 21mpd 15 1 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wal 1651   = wceq 1653  wex 1875  wnf 1879  wcel 2157  wnfc 2928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387
This theorem is referenced by:  vtocldf  3443  bj-vtoclgfALT  33513
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