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Theorem nfrald 2665
 Description: Deduction version of nfral 2667. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfrald.2 yφ
nfrald.3 (φxA)
nfrald.4 (φ → Ⅎxψ)
Assertion
Ref Expression
nfrald (φ → Ⅎxy A ψ)

Proof of Theorem nfrald
StepHypRef Expression
1 df-ral 2619 . 2 (y A ψy(y Aψ))
2 nfrald.2 . . 3 yφ
3 nfcvf 2511 . . . . . 6 x x = yxy)
43adantl 452 . . . . 5 ((φ ¬ x x = y) → xy)
5 nfrald.3 . . . . . 6 (φxA)
65adantr 451 . . . . 5 ((φ ¬ x x = y) → xA)
74, 6nfeld 2504 . . . 4 ((φ ¬ x x = y) → Ⅎx y A)
8 nfrald.4 . . . . 5 (φ → Ⅎxψ)
98adantr 451 . . . 4 ((φ ¬ x x = y) → Ⅎxψ)
107, 9nfimd 1808 . . 3 ((φ ¬ x x = y) → Ⅎx(y Aψ))
112, 10nfald2 1972 . 2 (φ → Ⅎxy(y Aψ))
121, 11nfxfrd 1571 1 (φ → Ⅎxy A ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by:  nfrexd  2666  nfral  2667
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