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Theorem dftr5 4677
Description: An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
Assertion
Ref Expression
dftr5 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dftr5
StepHypRef Expression
1 dftr2 4676 . 2 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2 alcom 2023 . . 3 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
3 impexp 460 . . . . . . . 8 (((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
43albii 1736 . . . . . . 7 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
5 df-ral 2900 . . . . . . 7 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑥𝐴𝑦𝐴)))
64, 5bitr4i 265 . . . . . 6 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑦𝑥 (𝑥𝐴𝑦𝐴))
7 r19.21v 2942 . . . . . 6 (∀𝑦𝑥 (𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
86, 7bitri 262 . . . . 5 (∀𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ (𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
98albii 1736 . . . 4 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
10 df-ral 2900 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑥 𝑦𝐴))
119, 10bitr4i 265 . . 3 (∀𝑥𝑦((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
122, 11bitri 262 . 2 (∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴) ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
131, 12bitri 262 1 (Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wcel 1976  wral 2895  Tr wtr 4674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-v 3174  df-in 3546  df-ss 3553  df-uni 4367  df-tr 4675
This theorem is referenced by:  dftr3  4678  smobeth  9264
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