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Theorem dvelimf 1933
Description: Version of dvelim 1935 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
Hypotheses
Ref Expression
dvelimf.1 (𝜑 → ∀𝑥𝜑)
dvelimf.2 (𝜓 → ∀𝑧𝜓)
dvelimf.3 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimf (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem dvelimf
StepHypRef Expression
1 dvelimf.1 . . 3 (𝜑 → ∀𝑥𝜑)
21hbsb4 1930 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → ∀𝑥[𝑦 / 𝑧]𝜑))
3 dvelimf.2 . . 3 (𝜓 → ∀𝑧𝜓)
4 dvelimf.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
53, 4sbieh 1714 . 2 ([𝑦 / 𝑧]𝜑𝜓)
65albii 1400 . 2 (∀𝑥[𝑦 / 𝑧]𝜑 ↔ ∀𝑥𝜓)
72, 5, 63imtr3g 202 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wal 1283  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  dvelim  1935  dveel1  1938  dveel2  1939
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