Theorem 3imtr4d 259

Description: More general version of 3imtr4i 257. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)

Hypotheses

Ref	Expression
3imtr4d.1	⊢ (φ → (ψ → χ))
3imtr4d.2	⊢ (φ → (θ ↔ ψ))
3imtr4d.3	⊢ (φ → (τ ↔ χ))

Assertion

Ref	Expression
3imtr4d	⊢ (φ → (θ → τ))

Proof of Theorem 3imtr4d

Step	Hyp	Ref	Expression
1		3imtr4d.2	. 2 ⊢ (φ → (θ ↔ ψ))
2		3imtr4d.1	. . 3 ⊢ (φ → (ψ → χ))
3		3imtr4d.3	. . 3 ⊢ (φ → (τ ↔ χ))
4	2, 3	sylibrd 225	. 2 ⊢ (φ → (ψ → τ))
5	1, 4	sylbid 206	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:  ax11indalem  2197  ax11inda2ALT  2198  leltfintr  4459  ltfintr  4460  ltfintri  4467  ltlefin  4469  tfinltfinlem1  4501  vfinspsslem1  4551  pw1fnf1o  5856  enprmaplem3  6079  nchoicelem9  6298