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Theorem animpimp2impd 844
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 457 . . . . 5 ((𝜓𝜑) → (𝜃 → (𝜂𝜏)))
43a2d 29 . . . 4 ((𝜓𝜑) → ((𝜃𝜂) → (𝜃𝜏)))
51, 4syld 47 . . 3 ((𝜓𝜑) → (𝜒 → (𝜃𝜏)))
65expcom 414 . 2 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
76a2d 29 1 (𝜑 → ((𝜓𝜒) → (𝜓 → (𝜃𝜏))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396
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 397
This theorem is referenced by:  seqcl2  13925  seqfveq2  13929  seqshft2  13933  monoord  13937  seqsplit  13940  seqid2  13953  seqhomo  13954  sylow1lem1  19378  imasdsf1olem  23724  ovolicc2lem3  24881  dvnres  25293  cvmliftlem7  33825  cvmliftlem10  33828  monoordxrv  43692  smonoord  45534
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