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Theorem dvelimf 1988
Description: Version of dvelim 1990 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 1985 . 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 1763 . 2 ([𝑦 / 𝑧]𝜑𝜓)
65albii 1446 . 2 (∀𝑥[𝑦 / 𝑧]𝜑 ↔ ∀𝑥𝜓)
72, 5, 63imtr3g 203 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wal 1329  [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  dvelim  1990  dveel1  1993  dveel2  1994
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