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Theorem fvmptdf 5160
 Description: Alternate deduction version of fvmpt 5151, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdf.1 (φA 𝐷)
fvmptdf.2 ((φ x = A) → B 𝑉)
fvmptdf.3 ((φ x = A) → ((𝐹A) = Bψ))
fvmptdf.4 x𝐹
fvmptdf.5 xψ
Assertion
Ref Expression
fvmptdf (φ → (𝐹 = (x 𝐷B) → ψ))
Distinct variable groups:   x,A   x,𝐷   φ,x
Allowed substitution hints:   ψ(x)   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmptdf
StepHypRef Expression
1 nfv 1404 . 2 xφ
2 fvmptdf.4 . . . 4 x𝐹
3 nfmpt1 3803 . . . 4 x(x 𝐷B)
42, 3nfeq 2168 . . 3 x 𝐹 = (x 𝐷B)
5 fvmptdf.5 . . 3 xψ
64, 5nfim 1448 . 2 x(𝐹 = (x 𝐷B) → ψ)
7 fvmptdf.1 . . . 4 (φA 𝐷)
8 elex 2542 . . . 4 (A 𝐷A V)
97, 8syl 14 . . 3 (φA V)
10 isset 2538 . . 3 (A V ↔ x x = A)
119, 10sylib 127 . 2 (φx x = A)
12 fveq1 5079 . . 3 (𝐹 = (x 𝐷B) → (𝐹A) = ((x 𝐷B)‘A))
13 simpr 103 . . . . . . 7 ((φ x = A) → x = A)
1413fveq2d 5084 . . . . . 6 ((φ x = A) → ((x 𝐷B)‘x) = ((x 𝐷B)‘A))
157adantr 261 . . . . . . . 8 ((φ x = A) → A 𝐷)
1613, 15eqeltrd 2097 . . . . . . 7 ((φ x = A) → x 𝐷)
17 fvmptdf.2 . . . . . . 7 ((φ x = A) → B 𝑉)
18 eqid 2023 . . . . . . . 8 (x 𝐷B) = (x 𝐷B)
1918fvmpt2 5156 . . . . . . 7 ((x 𝐷 B 𝑉) → ((x 𝐷B)‘x) = B)
2016, 17, 19syl2anc 393 . . . . . 6 ((φ x = A) → ((x 𝐷B)‘x) = B)
2114, 20eqtr3d 2057 . . . . 5 ((φ x = A) → ((x 𝐷B)‘A) = B)
2221eqeq2d 2034 . . . 4 ((φ x = A) → ((𝐹A) = ((x 𝐷B)‘A) ↔ (𝐹A) = B))
23 fvmptdf.3 . . . 4 ((φ x = A) → ((𝐹A) = Bψ))
2422, 23sylbid 139 . . 3 ((φ x = A) → ((𝐹A) = ((x 𝐷B)‘A) → ψ))
2512, 24syl5 28 . 2 ((φ x = A) → (𝐹 = (x 𝐷B) → ψ))
261, 6, 11, 25exlimdd 1736 1 (φ → (𝐹 = (x 𝐷B) → ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  Ⅎwnf 1329  ∃wex 1362   = wceq 1374   ∈ wcel 1376  Ⅎwnfc 2148  Vcvv 2534   ∈ cmpt 3771  ‘cfv 4806 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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-14 1388  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005  ax-sep 3828  ax-pow 3880  ax-pr 3897 This theorem depends on definitions:  df-bi 110  df-3an 878  df-tru 1232  df-nf 1330  df-sb 1629  df-eu 1885  df-mo 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-csb 2830  df-un 2901  df-in 2903  df-ss 2910  df-pw 3314  df-sn 3334  df-pr 3335  df-op 3337  df-uni 3534  df-br 3718  df-opab 3772  df-mpt 3773  df-id 3984  df-xp 4254  df-rel 4255  df-cnv 4256  df-co 4257  df-dm 4258  df-iota 4771  df-fun 4808  df-fv 4814 This theorem is referenced by:  fvmptdv  5161
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