Theorem 3imtr3d 202

Description: More general version of 3imtr3i 200. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)

Hypotheses

Ref	Expression
3imtr3d.1	⊢ (𝜑 → (𝜓 → 𝜒))
3imtr3d.2	⊢ (𝜑 → (𝜓 ↔ 𝜃))
3imtr3d.3	⊢ (𝜑 → (𝜒 ↔ 𝜏))

Assertion

Ref	Expression
3imtr3d	⊢ (𝜑 → (𝜃 → 𝜏))

Proof of Theorem 3imtr3d

Step	Hyp	Ref	Expression
1		3imtr3d.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
2		3imtr3d.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		3imtr3d.3	. . 3 ⊢ (𝜑 → (𝜒 ↔ 𝜏))
4	2, 3	sylibd 149	. 2 ⊢ (𝜑 → (𝜓 → 𝜏))
5	1, 4	sylbird 170	1 ⊢ (𝜑 → (𝜃 → 𝜏))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  f1imass  5824  focdmex  6181  tposfn2  6333  eroveu  6694  ismkvnex  7230  indpi  7428  axcaucvglemres  7985  qsqeqor  10761  caucvgrelemcau  11164  m1dvdsndvds  12444  pcpremul  12489  pcaddlem  12535  pockthlem  12552  issgrpd  13116  ghmf1  13481  islssmd  13993  znrrg  14294  limccnpcntop  15019  sincosq1sgn  15170  sincosq2sgn  15171  lgseisenlem2  15420  subctctexmid  15755  neap0mkv  15826