MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  animpimp2impd Structured version   Visualization version   GIF version

Theorem animpimp2impd 846
Description: Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
Hypotheses
Ref Expression
animpimp2impd.1 ((𝜓𝜑) → (𝜒 → (𝜃𝜂)))
animpimp2impd.2 ((𝜓 ∧ (𝜑𝜃)) → (𝜂𝜏))
Assertion
Ref Expression
animpimp2impd (𝜑 → ((𝜓𝜒) → (𝜓 → (𝜃𝜏))))

Proof of Theorem animpimp2impd
StepHypRef Expression
1 animpimp2impd.1 . . . 4 ((𝜓𝜑) → (𝜒 → (𝜃𝜂)))
2 animpimp2impd.2 . . . . . 6 ((𝜓 ∧ (𝜑𝜃)) → (𝜂𝜏))
32expr 460 . . . . 5 ((𝜓𝜑) → (𝜃 → (𝜂𝜏)))
43a2d 29 . . . 4 ((𝜓𝜑) → ((𝜃𝜂) → (𝜃𝜏)))
51, 4syld 47 . . 3 ((𝜓𝜑) → (𝜒 → (𝜃𝜏)))
65expcom 417 . 2 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
76a2d 29 1 (𝜑 → ((𝜓𝜒) → (𝜓 → (𝜃𝜏))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399
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 210  df-an 400
This theorem is referenced by:  seqcl2  13594  seqfveq2  13598  seqshft2  13602  monoord  13606  seqsplit  13609  seqid2  13622  seqhomo  13623  sylow1lem1  18987  imasdsf1olem  23271  ovolicc2lem3  24416  dvnres  24828  cvmliftlem7  32966  cvmliftlem10  32969  monoordxrv  42697  smonoord  44496
  Copyright terms: Public domain W3C validator