Theorem syl6rbb 186

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

Hypotheses

Ref	Expression
syl6rbb.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
syl6rbb.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
syl6rbb	⊢ (𝜑 → (𝜃 ↔ 𝜓))

Proof of Theorem syl6rbb

Step	Hyp	Ref	Expression
1		syl6rbb.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		syl6rbb.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
3	1, 2	syl6bb 185	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
4	3	bicomd 129	1 ⊢ (𝜑 → (𝜃 ↔ 𝜓))

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:  syl6rbbr  188  bibif  614  pm5.61  708  oranabs  728  pm5.7dc  861  nbbndc  1285  resopab2  4655  xpcom  4864  ac6sfi  6352  elznn0  8260  rexuz3  9588