Theorem sylbird 159

Description: A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Hypotheses

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

Assertion

Ref	Expression
sylbird	⊢ (φ → (ψ → θ))

Proof of Theorem sylbird

Step	Hyp	Ref	Expression
1		sylbird.1	. . 3 ⊢ (φ → (χ ↔ ψ))
2	1	biimprd 147	. 2 ⊢ (φ → (ψ → χ))
3		sylbird.2	. 2 ⊢ (φ → (χ → θ))
4	2, 3	syld 40	1 ⊢ (φ → (ψ → θ))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  3imtr3d  191  drex1  1653  eqreu  2702  onsucsssucr  4173  ordsucunielexmid  4189  ovi3  5549  tfrlem9  5846  rdgon  5882  distrlem4prl  6410  distrlem4pru  6411  recexprlemm  6446  aptisr  6516  renegcl  6872  remulext2  7188  mulext1  7200  apmul1  7350  nnge1  7521  0mnnnnn0  7789  nn0lt2  7888  zneo  7905  uzind2  7916  fzind  7919  nn0ind-raph  7921