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Theorem mpbiran3d 46120
Description: Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024.)
Hypotheses
Ref Expression
mpbiran3d.1 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
mpbiran3d.2 ((𝜑𝜒) → 𝜃)
Assertion
Ref Expression
mpbiran3d (𝜑 → (𝜓𝜒))

Proof of Theorem mpbiran3d
StepHypRef Expression
1 mpbiran3d.1 . . . 4 (𝜑 → (𝜓 ↔ (𝜒𝜃)))
21simprbda 499 . . 3 ((𝜑𝜓) → 𝜒)
32ex 413 . 2 (𝜑 → (𝜓𝜒))
4 mpbiran3d.2 . . . . 5 ((𝜑𝜒) → 𝜃)
54ex 413 . . . 4 (𝜑 → (𝜒𝜃))
65ancld 551 . . 3 (𝜑 → (𝜒 → (𝜒𝜃)))
76, 1sylibrd 258 . 2 (𝜑 → (𝜒𝜓))
83, 7impbid 211 1 (𝜑 → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  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:  mpbiran4d  46121  functhinc  46304  thincsect  46316  thincinv  46318  grptcmon  46355  grptcepi  46356
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