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Theorem 3imtr3d 195
Description: More general version of 3imtr3i 193. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
Hypotheses
Ref Expression
3imtr3d.1 (𝜑 → (𝜓𝜒))
3imtr3d.2 (𝜑 → (𝜓𝜃))
3imtr3d.3 (𝜑 → (𝜒𝜏))
Assertion
Ref Expression
3imtr3d (𝜑 → (𝜃𝜏))

Proof of Theorem 3imtr3d
StepHypRef Expression
1 3imtr3d.2 . 2 (𝜑 → (𝜓𝜃))
2 3imtr3d.1 . . 3 (𝜑 → (𝜓𝜒))
3 3imtr3d.3 . . 3 (𝜑 → (𝜒𝜏))
42, 3sylibd 142 . 2 (𝜑 → (𝜓𝜏))
51, 4sylbird 163 1 (𝜑 → (𝜃𝜏))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  f1imass  5440  fornex  5769  tposfn2  5911  freccl  6023  eroveu  6227  indpi  6497  axcaucvglemres  7030  caucvgrelemcau  9800
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