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  5980  focdmex  6344  tposfn2  6537  eroveu  6900  ismkvnex  7496  indpi  7710  axcaucvglemres  8267  qsqeqor  11101  caucvgrelemcau  11761  m1dvdsndvds  13049  pcpremul  13094  pcaddlem  13140  pockthlem  13157  issgrpd  13778  ghmf1  14127  islssmd  14747  znrrg  15046  limccnpcntop  15828  sincosq1sgn  15980  sincosq2sgn  15981  lgseisenlem2  16312  subctctexmid  17152  neap0mkv  17241