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  5953  focdmex  6317  tposfn2  6510  eroveu  6873  ismkvnex  7459  indpi  7673  axcaucvglemres  8230  qsqeqor  11039  caucvgrelemcau  11694  m1dvdsndvds  12975  pcpremul  13020  pcaddlem  13066  pockthlem  13083  issgrpd  13679  ghmf1  14030  islssmd  14637  znrrg  14938  limccnpcntop  15670  sincosq1sgn  15821  sincosq2sgn  15822  lgseisenlem2  16074  subctctexmid  16914  neap0mkv  16994