Theorem bitrd

Description: Deduction form of bitri . (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 14-Apr-2013)

	Ref	Expression
Hypotheses	bitrd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bitrd.2	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
Assertion	bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		bitrd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bitrd.2	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
3	1	pm5.74i	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) )
4	2	pm5.74i	⊢ ( ( 𝜑 → 𝜒 ) ↔ ( 𝜑 → 𝜃 ) )
5	3 4	bitri	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜃 ) )
6	5	pm5.74ri	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )