Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dford3lem1 Structured version   Visualization version   GIF version

Theorem dford3lem1 36611
Description: Lemma for dford3 36613. (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 4686 . . . . 5 (𝑦 = 𝑏 → (Tr 𝑦 ↔ Tr 𝑏))
21cbvralv 3147 . . . 4 (∀𝑦𝑁 Tr 𝑦 ↔ ∀𝑏𝑁 Tr 𝑏)
32biimpi 205 . . 3 (∀𝑦𝑁 Tr 𝑦 → ∀𝑏𝑁 Tr 𝑏)
43adantl 481 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 Tr 𝑏)
5 trss 4689 . . . . . 6 (Tr 𝑁 → (𝑏𝑁𝑏𝑁))
6 ssralv 3629 . . . . . 6 (𝑏𝑁 → (∀𝑦𝑁 Tr 𝑦 → ∀𝑦𝑏 Tr 𝑦))
75, 6syl6 34 . . . . 5 (Tr 𝑁 → (𝑏𝑁 → (∀𝑦𝑁 Tr 𝑦 → ∀𝑦𝑏 Tr 𝑦)))
87com23 84 . . . 4 (Tr 𝑁 → (∀𝑦𝑁 Tr 𝑦 → (𝑏𝑁 → ∀𝑦𝑏 Tr 𝑦)))
98imp 444 . . 3 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → (𝑏𝑁 → ∀𝑦𝑏 Tr 𝑦))
109ralrimiv 2948 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁𝑦𝑏 Tr 𝑦)
11 r19.26 3046 . 2 (∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) ↔ (∀𝑏𝑁 Tr 𝑏 ∧ ∀𝑏𝑁𝑦𝑏 Tr 𝑦))
124, 10, 11sylanbrc 695 1 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  wral 2896  wss 3540  Tr wtr 4680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-tr 4681
This theorem is referenced by:  dford3lem2  36612  dford3  36613
  Copyright terms: Public domain W3C validator