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Theorem dvelimf 1939
Description: Version of dvelim 1941 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 1936 . 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 1720 . 2 ([𝑦 / 𝑧]𝜑𝜓)
65albii 1404 . 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 1287  [wsb 1692
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 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  dvelim  1941  dveel1  1944  dveel2  1945
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