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Theorem vtoclgft 2669
Description: Closed theorem form of vtoclgf 2677. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
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 elex 2630 . 2 (𝐴𝑉𝐴 ∈ V)
2 elisset 2633 . . . . 5 (𝐴 ∈ V → ∃𝑧 𝑧 = 𝐴)
323ad2ant3 966 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑧 𝑧 = 𝐴)
4 nfnfc1 2231 . . . . . . 7 𝑥𝑥𝐴
5 nfcvd 2229 . . . . . . . 8 (𝑥𝐴𝑥𝑧)
6 id 19 . . . . . . . 8 (𝑥𝐴𝑥𝐴)
75, 6nfeqd 2243 . . . . . . 7 (𝑥𝐴 → Ⅎ𝑥 𝑧 = 𝐴)
8 eqeq1 2094 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
98a1i 9 . . . . . . 7 (𝑥𝐴 → (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴)))
104, 7, 9cbvexd 1850 . . . . . 6 (𝑥𝐴 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
1110ad2antrr 472 . . . . 5 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
12113adant3 963 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
133, 12mpbid 145 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑥 𝑥 = 𝐴)
14 bi1 116 . . . . . . . . 9 ((𝜑𝜓) → (𝜑𝜓))
1514imim2i 12 . . . . . . . 8 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
1615com23 77 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜑 → (𝑥 = 𝐴𝜓)))
1716imp 122 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
1817alanimi 1393 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴𝜓))
19183ad2ant2 965 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∀𝑥(𝑥 = 𝐴𝜓))
20 simp1r 968 . . . . 5 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → Ⅎ𝑥𝜓)
21 19.23t 1612 . . . . 5 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
2220, 21syl 14 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
2319, 22mpbid 145 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∃𝑥 𝑥 = 𝐴𝜓))
2413, 23mpd 13 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → 𝜓)
251, 24syl3an3 1209 1 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924  wal 1287   = wceq 1289  wnf 1394  wex 1426  wcel 1438  wnfc 2215  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  vtocldf  2670
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