MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.12vv Structured version   Visualization version   GIF version

Theorem 19.12vv 2337
Description: Special case of 19.12 2315 where its converse holds. See 19.12vvv 1984 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
19.12vv (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem 19.12vv
StepHypRef Expression
1 19.21v 1934 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21exbii 1842 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
3 nfv 1909 . . . 4 𝑥𝜓
43nfal 2311 . . 3 𝑥𝑦𝜓
5419.36 2218 . 2 (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
6 19.36v 1983 . . . 4 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
76albii 1813 . . 3 (∀𝑦𝑥(𝜑𝜓) ↔ ∀𝑦(∀𝑥𝜑𝜓))
8 nfv 1909 . . . . 5 𝑦𝜑
98nfal 2311 . . . 4 𝑦𝑥𝜑
10919.21 2195 . . 3 (∀𝑦(∀𝑥𝜑𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
117, 10bitr2i 275 . 2 ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
122, 5, 113bitri 296 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wex 1773
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-10 2129  ax-11 2146  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator