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Theorem dford3lem1 37412
Description: Lemma for dford3 37414. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
dford3lem1 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
Distinct variable group:   𝑦,𝑏,𝑁

Proof of Theorem dford3lem1
StepHypRef Expression
1 treq 4749 . . . . 5 (𝑦 = 𝑏 → (Tr 𝑦 ↔ Tr 𝑏))
21cbvralv 3166 . . . 4 (∀𝑦𝑁 Tr 𝑦 ↔ ∀𝑏𝑁 Tr 𝑏)
32biimpi 206 . . 3 (∀𝑦𝑁 Tr 𝑦 → ∀𝑏𝑁 Tr 𝑏)
43adantl 482 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 Tr 𝑏)
5 trss 4752 . . . . . 6 (Tr 𝑁 → (𝑏𝑁𝑏𝑁))
6 ssralv 3658 . . . . . 6 (𝑏𝑁 → (∀𝑦𝑁 Tr 𝑦 → ∀𝑦𝑏 Tr 𝑦))
75, 6syl6 35 . . . . 5 (Tr 𝑁 → (𝑏𝑁 → (∀𝑦𝑁 Tr 𝑦 → ∀𝑦𝑏 Tr 𝑦)))
87com23 86 . . . 4 (Tr 𝑁 → (∀𝑦𝑁 Tr 𝑦 → (𝑏𝑁 → ∀𝑦𝑏 Tr 𝑦)))
98imp 445 . . 3 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → (𝑏𝑁 → ∀𝑦𝑏 Tr 𝑦))
109ralrimiv 2962 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁𝑦𝑏 Tr 𝑦)
11 r19.26 3060 . 2 (∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) ↔ (∀𝑏𝑁 Tr 𝑏 ∧ ∀𝑏𝑁𝑦𝑏 Tr 𝑦))
124, 10, 11sylanbrc 697 1 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  wral 2909  wss 3567  Tr wtr 4743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428  df-tr 4744
This theorem is referenced by:  dford3lem2  37413  dford3  37414
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