Theorem bitr2di 197

Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
bitr2di.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bitr2di.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
bitr2di	⊢ (𝜑 → (𝜃 ↔ 𝜓))

Proof of Theorem bitr2di

Step	Hyp	Ref	Expression
1		bitr2di.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		bitr2di.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
3	1, 2	bitrdi 196	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
4	3	bicomd 141	1 ⊢ (𝜑 → (𝜃 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bitr4id  199  bibif  710  pm5.61  806  oranabs  827  pm5.7dc  967  nbbndc  1443  resopab2  5108  xpcom  5332  f1od2  6465  map1  7095  ac6sfi  7196  elznn0  9642  rexuz3  11739  xrmaxiflemcom  11998  metrest  15590  sincosq3sgn  15912  sincosq4sgn  15913  lgsquadlem3  16181  pw1map  17008