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

Theorem vtoclgft 2708
Description: Closed theorem form of vtoclgf 2716. (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 2669 . 2 (𝐴𝑉𝐴 ∈ V)
2 elisset 2672 . . . . 5 (𝐴 ∈ V → ∃𝑧 𝑧 = 𝐴)
323ad2ant3 987 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑧 𝑧 = 𝐴)
4 nfnfc1 2259 . . . . . . 7 𝑥𝑥𝐴
5 nfcvd 2257 . . . . . . . 8 (𝑥𝐴𝑥𝑧)
6 id 19 . . . . . . . 8 (𝑥𝐴𝑥𝐴)
75, 6nfeqd 2271 . . . . . . 7 (𝑥𝐴 → Ⅎ𝑥 𝑧 = 𝐴)
8 eqeq1 2122 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
98a1i 9 . . . . . . 7 (𝑥𝐴 → (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴)))
104, 7, 9cbvexd 1877 . . . . . 6 (𝑥𝐴 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
1110ad2antrr 477 . . . . 5 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
12113adant3 984 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
133, 12mpbid 146 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∃𝑥 𝑥 = 𝐴)
14 bi1 117 . . . . . . . . 9 ((𝜑𝜓) → (𝜑𝜓))
1514imim2i 12 . . . . . . . 8 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
1615com23 78 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜑 → (𝑥 = 𝐴𝜓)))
1716imp 123 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
1817alanimi 1418 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴𝜓))
19183ad2ant2 986 . . . 4 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → ∀𝑥(𝑥 = 𝐴𝜓))
20 simp1r 989 . . . . 5 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ V) → Ⅎ𝑥𝜓)
21 19.23t 1638 . . . . 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 146 . . 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 1234 1 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945  wal 1312   = wceq 1314  wnf 1419  wex 1451  wcel 1463  wnfc 2243  Vcvv 2658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by:  vtocldf  2709
  Copyright terms: Public domain W3C validator