Theorem 3imtr3i 200

Description: A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)

Hypotheses

Ref	Expression
3imtr3.1	⊢ (𝜑 → 𝜓)
3imtr3.2	⊢ (𝜑 ↔ 𝜒)
3imtr3.3	⊢ (𝜓 ↔ 𝜃)

Assertion

Ref	Expression
3imtr3i	⊢ (𝜒 → 𝜃)

Proof of Theorem 3imtr3i

Step	Hyp	Ref	Expression
1		3imtr3.2	. . 3 ⊢ (𝜑 ↔ 𝜒)
2		3imtr3.1	. . 3 ⊢ (𝜑 → 𝜓)
3	1, 2	sylbir 135	. 2 ⊢ (𝜒 → 𝜓)
4		3imtr3.3	. 2 ⊢ (𝜓 ↔ 𝜃)
5	3, 4	sylib 122	1 ⊢ (𝜒 → 𝜃)

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:  cbv1  1745  cbv1v  1747  moimv  2092  hblem  2285  tfi  4583  smores  6295  idssen  6779  suplocsrlem  7809  bezoutlemle  12011  limcmpted  14171  sincosq3sgn  14288  subctctexmid  14789  dcapnconstALT  14849