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Theorem dvelimf 2027
Description: Version of dvelim 2029 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 2024 . 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 1801 . 2 ([𝑦 / 𝑧]𝜑𝜓)
65albii 1481 . 2 (∀𝑥[𝑦 / 𝑧]𝜑 ↔ ∀𝑥𝜓)
72, 5, 63imtr3g 204 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wal 1362  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  dvelim  2029  dveel1  2169  dveel2  2170
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