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Theorem dford3lem1 42343
Description: Lemma for dford3 42345. (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 5266 . . . . 5 (𝑦 = 𝑏 → (Tr 𝑦 ↔ Tr 𝑏))
21cbvralvw 3228 . . . 4 (∀𝑦𝑁 Tr 𝑦 ↔ ∀𝑏𝑁 Tr 𝑏)
32biimpi 215 . . 3 (∀𝑦𝑁 Tr 𝑦 → ∀𝑏𝑁 Tr 𝑏)
43adantl 481 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 Tr 𝑏)
5 trss 5269 . . . . . 6 (Tr 𝑁 → (𝑏𝑁𝑏𝑁))
6 ssralv 4045 . . . . . 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 406 . . 3 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → (𝑏𝑁 → ∀𝑦𝑏 Tr 𝑦))
109ralrimiv 3139 . 2 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁𝑦𝑏 Tr 𝑦)
11 r19.26 3105 . 2 (∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) ↔ (∀𝑏𝑁 Tr 𝑏 ∧ ∀𝑏𝑁𝑦𝑏 Tr 𝑦))
124, 10, 11sylanbrc 582 1 ((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wral 3055  wss 3943  Tr wtr 5258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-v 3470  df-in 3950  df-ss 3960  df-uni 4903  df-tr 5259
This theorem is referenced by:  dford3lem2  42344  dford3  42345
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