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Theorem dvelimfALT2 1789
 Description: Proof of dvelimf 1990 using dveeq2 1787 (shown as the last hypothesis) instead of ax-12 1489. This shows that ax-12 1489 could be replaced by dveeq2 1787 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)
Hypotheses
Ref Expression
dvelimfALT2.1 (𝜑 → ∀𝑥𝜑)
dvelimfALT2.2 (𝜓 → ∀𝑧𝜓)
dvelimfALT2.3 (𝑧 = 𝑦 → (𝜑𝜓))
dvelimfALT2.4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Assertion
Ref Expression
dvelimfALT2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem dvelimfALT2
StepHypRef Expression
1 ax-17 1506 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2 hbn1 1630 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
3 dvelimfALT2.4 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
4 dvelimfALT2.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
54a1i 9 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
62, 3, 5hbimd 1552 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑦𝜑) → ∀𝑥(𝑧 = 𝑦𝜑)))
71, 6hbald 1467 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑)))
8 dvelimfALT2.2 . . 3 (𝜓 → ∀𝑧𝜓)
9 dvelimfALT2.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
108, 9equsalh 1704 . 2 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
1110albii 1446 . 2 (∀𝑥𝑧(𝑧 = 𝑦𝜑) ↔ ∀𝑥𝜓)
127, 10, 113imtr3g 203 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331 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-in1 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337 This theorem is referenced by: (None)
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