ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtoclgft GIF version

Theorem vtoclgft 2621
Description: Closed theorem form of vtoclgf 2629. (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 2583 . 2 (𝐴𝑉𝐴 ∈ V)
2 elisset 2585 . . . . 5 (𝐴 ∈ V → ∃𝑧 𝑧 = 𝐴)
323ad2ant3 938 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑧 𝑧 = 𝐴)
4 nfnfc1 2197 . . . . . . 7 𝑥𝑥𝐴
5 nfcvd 2195 . . . . . . . 8 (𝑥𝐴𝑥𝑧)
6 id 19 . . . . . . . 8 (𝑥𝐴𝑥𝐴)
75, 6nfeqd 2208 . . . . . . 7 (𝑥𝐴 → Ⅎ𝑥 𝑧 = 𝐴)
8 eqeq1 2062 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
98a1i 9 . . . . . . 7 (𝑥𝐴 → (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴)))
104, 7, 9cbvexd 1818 . . . . . 6 (𝑥𝐴 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
1110ad2antrr 465 . . . . 5 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
12113adant3 935 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
133, 12mpbid 139 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑥 𝑥 = 𝐴)
14 bi1 115 . . . . . . . . 9 ((𝜑𝜓) → (𝜑𝜓))
1514imim2i 12 . . . . . . . 8 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
1615com23 76 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜑 → (𝑥 = 𝐴𝜓)))
1716imp 119 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
1817alanimi 1364 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴𝜓))
19183ad2ant2 937 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∀𝑥(𝑥 = 𝐴𝜓))
20 simp1r 940 . . . . 5 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → Ⅎ𝑥𝜓)
21 19.23t 1583 . . . . 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 139 . . 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 1181 1 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896  wal 1257   = wceq 1259  wnf 1365  wex 1397  wcel 1409  wnfc 2181  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  vtocldf  2622
  Copyright terms: Public domain W3C validator