MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trintOLD Structured version   Visualization version   GIF version

Theorem trintOLD 4960
Description: Obsolete proof of trintOLD 4960 as of 3-Oct-2022. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintOLD (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem trintOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dftr3 4947 . . . . 5 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
21ralbii 3159 . . . 4 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝑥)
3 df-ral 3092 . . . . . 6 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦𝑥))
43ralbii 3159 . . . . 5 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥))
5 ralcom4 3410 . . . . 5 (∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
64, 5bitri 267 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
72, 6sylbb 211 . . 3 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
8 ralim 3127 . . 3 (∀𝑥𝐴 (𝑦𝑥𝑦𝑥) → (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
97, 8sylg 1918 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
10 dftr3 4947 . . 3 (Tr 𝐴 ↔ ∀𝑦 𝐴𝑦 𝐴)
11 df-ral 3092 . . . 4 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(𝑦 𝐴𝑦 𝐴))
12 vex 3386 . . . . . . 7 𝑦 ∈ V
1312elint2 4672 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
14 ssint 4681 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
1513, 14imbi12i 342 . . . . 5 ((𝑦 𝐴𝑦 𝐴) ↔ (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1615albii 1915 . . . 4 (∀𝑦(𝑦 𝐴𝑦 𝐴) ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1711, 16bitri 267 . . 3 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1810, 17bitri 267 . 2 (Tr 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
199, 18sylibr 226 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651  wcel 2157  wral 3087  wss 3767   cint 4665  Tr wtr 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-v 3385  df-in 3774  df-ss 3781  df-uni 4627  df-int 4666  df-tr 4944
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator