Theorem 3imtr3i 256

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 204	. 2 ⊢ (χ → ψ)
4		3imtr3.3	. 2 ⊢ (ψ ↔ θ)
5	3, 4	sylib 188	1 ⊢ (χ → θ)

Syntax hints:   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  rb-ax1  1517  speimfw  1645  19.8wOLD  1693  ax10lem2  1937  sbal  2127  hblem  2458  fvfullfunlem2  5863  idssen  6041