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Theorem 3imtr3d 201
 Description: More general version of 3imtr3i 199. 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 148 . 2 (𝜑 → (𝜓𝜏))
51, 4sylbird 169 1 (𝜑 → (𝜃𝜏))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  f1imass  5641  fornex  5979  tposfn2  6129  eroveu  6486  ismkvnex  6995  indpi  7114  axcaucvglemres  7671  caucvgrelemcau  10703  limccnpcntop  12719  subctctexmid  13030
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