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  5818  focdmex  6169  tposfn2  6321  eroveu  6682  ismkvnex  7216  indpi  7404  axcaucvglemres  7961  qsqeqor  10724  caucvgrelemcau  11127  m1dvdsndvds  12389  pcpremul  12434  pcaddlem  12480  pockthlem  12497  issgrpd  12998  ghmf1  13346  islssmd  13858  znrrg  14159  limccnpcntop  14854  sincosq1sgn  15002  sincosq2sgn  15003  lgseisenlem2  15228  subctctexmid  15561  neap0mkv  15629