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Theorem dvelimfALT2 1714
Description: Proof of dvelimf 1907 using dveeq2 1712 (shown as the last hypothesis) instead of ax-12 1418. This shows that ax-12 1418 could be replaced by dveeq2 1712 (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 1435 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2 hbn1 1558 . . . 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 1481 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑦𝜑) → ∀𝑥(𝑧 = 𝑦𝜑)))
71, 6hbald 1396 . 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 1630 . 2 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
1110albii 1375 . 2 (∀𝑥𝑧(𝑧 = 𝑦𝜑) ↔ ∀𝑥𝜓)
127, 10, 113imtr3g 197 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wal 1257   = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265
This theorem is referenced by: (None)
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