Theorem 3imtr3g 260

Description: More general version of 3imtr3i 256. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Hypotheses

Ref	Expression
3imtr3g.1	⊢ (φ → (ψ → χ))
3imtr3g.2	⊢ (ψ ↔ θ)
3imtr3g.3	⊢ (χ ↔ τ)

Assertion

Ref	Expression
3imtr3g	⊢ (φ → (θ → τ))

Proof of Theorem 3imtr3g

Step	Hyp	Ref	Expression
1		3imtr3g.2	. . 3 ⊢ (ψ ↔ θ)
2		3imtr3g.1	. . 3 ⊢ (φ → (ψ → χ))
3	1, 2	syl5bir 209	. 2 ⊢ (φ → (θ → χ))
4		3imtr3g.3	. 2 ⊢ (χ ↔ τ)
5	3, 4	syl6ib 217	1 ⊢ (φ → (θ → τ))

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:  exim  1575  dvelimhw  1849  ax12olem2  1928  dvelimh  1964  dvelimALT  2133  dvelimf-o  2180  axi11e  2332  sspwb  4119  pw1disj  4168  ssopab2b  4714  imadif  5172  enpw1  6063