MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.36i Structured version   Visualization version   GIF version

Theorem 19.36i 2097
Description: Inference associated with 19.36 2096. See 19.36iv 1902 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
Hypotheses
Ref Expression
19.36.1 𝑥𝜓
19.36i.2 𝑥(𝜑𝜓)
Assertion
Ref Expression
19.36i (∀𝑥𝜑𝜓)

Proof of Theorem 19.36i
StepHypRef Expression
1 19.36i.2 . 2 𝑥(𝜑𝜓)
2 19.36.1 . . 3 𝑥𝜓
3219.36 2096 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3mpbi 220 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707
This theorem is referenced by:  spimv1  2112  spim  2253  vtoclf  3244  bj-vtoclf  32552
  Copyright terms: Public domain W3C validator