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Theorem vtoclgft 2598
Description: Closed theorem form of vtoclgf 2606. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
vtoclgft (((xA xψ) (x(x = A → (φψ)) xφ) A 𝑉) → ψ)

Proof of Theorem vtoclgft
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 elex 2560 . 2 (A 𝑉A V)
2 elisset 2562 . . . . 5 (A V → z z = A)
323ad2ant3 926 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → z z = A)
4 nfnfc1 2178 . . . . . . 7 xxA
5 nfcvd 2176 . . . . . . . 8 (xAxz)
6 id 19 . . . . . . . 8 (xAxA)
75, 6nfeqd 2189 . . . . . . 7 (xA → Ⅎx z = A)
8 eqeq1 2043 . . . . . . . 8 (z = x → (z = Ax = A))
98a1i 9 . . . . . . 7 (xA → (z = x → (z = Ax = A)))
104, 7, 9cbvexd 1799 . . . . . 6 (xA → (z z = Ax x = A))
1110ad2antrr 457 . . . . 5 (((xA xψ) (x(x = A → (φψ)) xφ)) → (z z = Ax x = A))
12113adant3 923 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → (z z = Ax x = A))
133, 12mpbid 135 . . 3 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → x x = A)
14 bi1 111 . . . . . . . . 9 ((φψ) → (φψ))
1514imim2i 12 . . . . . . . 8 ((x = A → (φψ)) → (x = A → (φψ)))
1615com23 72 . . . . . . 7 ((x = A → (φψ)) → (φ → (x = Aψ)))
1716imp 115 . . . . . 6 (((x = A → (φψ)) φ) → (x = Aψ))
1817alanimi 1345 . . . . 5 ((x(x = A → (φψ)) xφ) → x(x = Aψ))
19183ad2ant2 925 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → x(x = Aψ))
20 simp1r 928 . . . . 5 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → Ⅎxψ)
21 19.23t 1564 . . . . 5 (Ⅎxψ → (x(x = Aψ) ↔ (x x = Aψ)))
2220, 21syl 14 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → (x(x = Aψ) ↔ (x x = Aψ)))
2319, 22mpbid 135 . . 3 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → (x x = Aψ))
2413, 23mpd 13 . 2 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → ψ)
251, 24syl3an3 1169 1 (((xA xψ) (x(x = A → (φψ)) xφ) A 𝑉) → ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884  wal 1240   = wceq 1242  wnf 1346  wex 1378   wcel 1390  wnfc 2162  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  vtocldf  2599
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