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

Theorem aaanOLD 2329
Description: Obsolete version of aaan 2328 as of 21-Nov-2024. (Contributed by NM, 12-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
aaan.1 𝑦𝜑
aaan.2 𝑥𝜓
Assertion
Ref Expression
aaanOLD (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Proof of Theorem aaanOLD
StepHypRef Expression
1 aaan.1 . . . 4 𝑦𝜑
2119.28 2222 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
32albii 1822 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4 aaan.2 . . . 4 𝑥𝜓
54nfal 2317 . . 3 𝑥𝑦𝜓
6519.27 2221 . 2 (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
73, 6bitri 275 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wal 1540  wnf 1786
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-10 2138  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator