Theorem 3bitrd 422

Description: Deduction from transitivity of biconditional.

Hypotheses

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

Assertion

Ref	Expression
3bitrd	⊢ (φ → (ψ ↔ τ))

Proof of Theorem 3bitrd

Step	Hyp	Ref	Expression
1		3bitrd.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		3bitrd.2	. . 3 ⊢ (φ → (χ ↔ θ))
3	1, 2	bitrd 406	. 2 ⊢ (φ → (ψ ↔ θ))
4		3bitrd.3	. 2 ⊢ (φ → (θ ↔ τ))
5	3, 4	bitrd 406	1 ⊢ (φ → (ψ ↔ τ))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  sbcgf 1469  dedth3v 1786  elimhyp3v 1792  keephyp3v 1795  unfilem3 3440  r1pwcl 3530  atcv0eq 5767

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org