MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  exp42 Structured version   Visualization version   GIF version

Theorem exp42 437
Description: An exportation inference. (Contributed by NM, 26-Apr-1994.)
Hypothesis
Ref Expression
exp42.1 (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
exp42 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Proof of Theorem exp42
StepHypRef Expression
1 exp42.1 . . 3 (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)
21exp31 421 . 2 (𝜑 → ((𝜓𝜒) → (𝜃𝜏)))
32expd 417 1 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 206  df-an 398
This theorem is referenced by:  isofrlem  7324  f1ocnv2d  7646  oelim  8521  zorn2lem7  10484  addrid  11381  initoeu1  17948  termoeu1  17955  issubg4  19010  lmodvsdir  20473  lmodvsass  20474  gsummatr01lem4  22129  dvfsumrlim3  25519  wwlksext2clwwlk  29277  shscli  30535  f1o3d  31820  slmdvsdir  32332  slmdvsass  32333  lshpcmp  37764
  Copyright terms: Public domain W3C validator