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Theorem vtoclgft 2581
Description: Closed theorem form of vtoclgf 2589. (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 2543 . 2 (A 𝑉A V)
2 elisset 2545 . . . . 5 (A V → z z = A)
323ad2ant3 915 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → z z = A)
4 nfnfc1 2163 . . . . . . 7 xxA
5 nfcvd 2161 . . . . . . . 8 (xAxz)
6 id 19 . . . . . . . 8 (xAxA)
75, 6nfeqd 2174 . . . . . . 7 (xA → Ⅎx z = A)
8 eqeq1 2028 . . . . . . . 8 (z = x → (z = Ax = A))
98a1i 9 . . . . . . 7 (xA → (z = x → (z = Ax = A)))
104, 7, 9cbvexd 1784 . . . . . 6 (xA → (z z = Ax x = A))
1110ad2antrr 460 . . . . 5 (((xA xψ) (x(x = A → (φψ)) xφ)) → (z z = Ax x = A))
12113adant3 912 . . . 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 1328 . . . . 5 ((x(x = A → (φψ)) xφ) → x(x = Aψ))
19183ad2ant2 914 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → x(x = Aψ))
20 simp1r 917 . . . . 5 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → Ⅎxψ)
21 19.23t 1549 . . . . 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 1156 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 873  wal 1226   = wceq 1228  wnf 1329  wex 1362   wcel 1374  wnfc 2147  Vcvv 2535
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537
This theorem is referenced by:  vtocldf  2582
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