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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  f1imass  5983  focdmex  6348  tposfn2  6542  eroveu  6905  ismkvnex  7502  indpi  7716  axcaucvglemres  8273  qsqeqor  11108  caucvgrelemcau  11768  m1dvdsndvds  13056  pcpremul  13101  pcaddlem  13147  pockthlem  13164  issgrpd  13786  ghmf1  14135  islssmd  14805  znrrg  15104  limccnpcntop  15934  sincosq1sgn  16086  sincosq2sgn  16087  lgseisenlem2  16428  subctctexmid  17268  neap0mkv  17359