Theorem syl6bir 162

Description: A mixed syllogism inference. (Contributed by NM, 18-May-1994.)

Hypotheses

Ref	Expression
syl6bir.1	⊢ (𝜑 → (𝜒 ↔ 𝜓))
syl6bir.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
syl6bir	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem syl6bir

Step	Hyp	Ref	Expression
1		syl6bir.1	. . 3 ⊢ (𝜑 → (𝜒 ↔ 𝜓))
2	1	biimprd 156	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		syl6bir.2	. 2 ⊢ (𝜒 → 𝜃)
4	2, 3	syl6 33	1 ⊢ (𝜑 → (𝜓 → 𝜃))

Syntax hints:   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  exdistrfor  1721  cbvexdh  1842  repizf2  3936  issref  4727  fnun  5025  ovigg  5641  tfrlem9  5958  tfri3  5976  ordge1n0im  6042  nntri3or  6095  axprecex  7046  peano5nnnn  7058  peano5nni  8042  zeo  8452  nn0ind-raph  8464  fzm1  9117  fzind2  9248  fzfig  9422  bcpasc  9693  climrecvg1n  10185  oddnn02np1  10280  oddge22np1  10281  evennn02n  10282  evennn2n  10283  gcdaddm  10375  coprmdvds1  10473  qredeq  10478  bj-intabssel  10599