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Theorem dvelimdf 2082
 Description: Deduction form of dvelimf 1997. This version may be useful if we want to avoid ax-17 1616 and use ax-16 2144 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
dvelimdf.1 xφ
dvelimdf.2 zφ
dvelimdf.3 (φ → Ⅎxψ)
dvelimdf.4 (φ → Ⅎzχ)
dvelimdf.5 (φ → (z = y → (ψχ)))
Assertion
Ref Expression
dvelimdf (φ → (¬ x x = y → Ⅎxχ))

Proof of Theorem dvelimdf
StepHypRef Expression
1 dvelimdf.2 . . . . . 6 zφ
2 dvelimdf.3 . . . . . 6 (φ → Ⅎxψ)
31, 2alrimi 1765 . . . . 5 (φzxψ)
4 nfsb4t 2080 . . . . 5 (zxψ → (¬ x x = y → Ⅎx[y / z]ψ))
53, 4syl 15 . . . 4 (φ → (¬ x x = y → Ⅎx[y / z]ψ))
65imp 418 . . 3 ((φ ¬ x x = y) → Ⅎx[y / z]ψ)
7 dvelimdf.1 . . . . 5 xφ
8 nfnae 1956 . . . . 5 x ¬ x x = y
97, 8nfan 1824 . . . 4 x(φ ¬ x x = y)
10 dvelimdf.4 . . . . . 6 (φ → Ⅎzχ)
11 dvelimdf.5 . . . . . 6 (φ → (z = y → (ψχ)))
121, 10, 11sbied 2036 . . . . 5 (φ → ([y / z]ψχ))
1312adantr 451 . . . 4 ((φ ¬ x x = y) → ([y / z]ψχ))
149, 13nfbidf 1774 . . 3 ((φ ¬ x x = y) → (Ⅎx[y / z]ψ ↔ Ⅎxχ))
156, 14mpbid 201 . 2 ((φ ¬ x x = y) → Ⅎxχ)
1615ex 423 1 (φ → (¬ x x = y → Ⅎxχ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  dvelimdc  2509
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