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Theorem nfreud 2783
 Description: Deduction version of nfreu 2785. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfreud.1 yφ
nfreud.2 (φxA)
nfreud.3 (φ → Ⅎxψ)
Assertion
Ref Expression
nfreud (φ → Ⅎx∃!y A ψ)

Proof of Theorem nfreud
StepHypRef Expression
1 df-reu 2621 . 2 (∃!y A ψ∃!y(y A ψ))
2 nfreud.1 . . 3 yφ
3 nfcvf 2511 . . . . . 6 x x = yxy)
43adantl 452 . . . . 5 ((φ ¬ x x = y) → xy)
5 nfreud.2 . . . . . 6 (φxA)
65adantr 451 . . . . 5 ((φ ¬ x x = y) → xA)
74, 6nfeld 2504 . . . 4 ((φ ¬ x x = y) → Ⅎx y A)
8 nfreud.3 . . . . 5 (φ → Ⅎxψ)
98adantr 451 . . . 4 ((φ ¬ x x = y) → Ⅎxψ)
107, 9nfand 1822 . . 3 ((φ ¬ x x = y) → Ⅎx(y A ψ))
112, 10nfeud2 2216 . 2 (φ → Ⅎx∃!y(y A ψ))
121, 11nfxfrd 1571 1 (φ → Ⅎx∃!y A ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Ⅎwnfc 2476  ∃!wreu 2616 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-nfc 2478  df-reu 2621 This theorem is referenced by:  nfreu  2785
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