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Theorem vtoclgft 2905
 Description: Closed theorem form of vtoclgf 2913. (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 2867 . 2 (A VA V)
2 elisset 2869 . . . . 5 (A V → z z = A)
323ad2ant3 978 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → z z = A)
4 nfnfc1 2492 . . . . . . 7 xxA
5 nfcvd 2490 . . . . . . . 8 (xAxz)
6 id 19 . . . . . . . 8 (xAxA)
75, 6nfeqd 2503 . . . . . . 7 (xA → Ⅎx z = A)
8 eqeq1 2359 . . . . . . . 8 (z = x → (z = Ax = A))
98a1i 10 . . . . . . 7 (xA → (z = x → (z = Ax = A)))
104, 7, 9cbvexd 2009 . . . . . 6 (xA → (z z = Ax x = A))
1110ad2antrr 706 . . . . 5 (((xA xψ) (x(x = A → (φψ)) xφ)) → (z z = Ax x = A))
12113adant3 975 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → (z z = Ax x = A))
133, 12mpbid 201 . . 3 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → x x = A)
14 bi1 178 . . . . . . . . 9 ((φψ) → (φψ))
1514imim2i 13 . . . . . . . 8 ((x = A → (φψ)) → (x = A → (φψ)))
1615com23 72 . . . . . . 7 ((x = A → (φψ)) → (φ → (x = Aψ)))
1716imp 418 . . . . . 6 (((x = A → (φψ)) φ) → (x = Aψ))
1817alanimi 1562 . . . . 5 ((x(x = A → (φψ)) xφ) → x(x = Aψ))
19183ad2ant2 977 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → x(x = Aψ))
20 simp1r 980 . . . . 5 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → Ⅎxψ)
21 19.23t 1800 . . . . 5 (Ⅎxψ → (x(x = Aψ) ↔ (x x = Aψ)))
2220, 21syl 15 . . . 4 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → (x(x = Aψ) ↔ (x x = Aψ)))
2319, 22mpbid 201 . . 3 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → (x x = Aψ))
2413, 23mpd 14 . 2 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → ψ)
251, 24syl3an3 1217 1 (((xA xψ) (x(x = A → (φψ)) xφ) A V) → ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by:  vtocldf  2906  iota2df  4365
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