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Theorem dford3lem1 41393
Description: Lemma for dford3 41395. (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 5231 . . . . 5 (𝑦 = 𝑏 → (Tr 𝑦 ↔ Tr 𝑏))
21cbvralvw 3224 . . . 4 (∀𝑦𝑁 Tr 𝑦 ↔ ∀𝑏𝑁 Tr 𝑏)
32biimpi 215 . . 3 (∀𝑦𝑁 Tr 𝑦 → ∀𝑏𝑁 Tr 𝑏)
43adantl 483 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 Tr 𝑏)
5 trss 5234 . . . . . 6 (Tr 𝑁 → (𝑏𝑁𝑏𝑁))
6 ssralv 4011 . . . . . 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 408 . . 3 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → (𝑏𝑁 → ∀𝑦𝑏 Tr 𝑦))
109ralrimiv 3139 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁𝑦𝑏 Tr 𝑦)
11 r19.26 3111 . 2 (∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) ↔ (∀𝑏𝑁 Tr 𝑏 ∧ ∀𝑏𝑁𝑦𝑏 Tr 𝑦))
124, 10, 11sylanbrc 584 1 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  wss 3911  Tr wtr 5223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3446  df-in 3918  df-ss 3928  df-uni 4867  df-tr 5224
This theorem is referenced by:  dford3lem2  41394  dford3  41395
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