Theorem bibi1d

Description: Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993)

	Ref	Expression
Hypothesis	imbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	bibi1d	⊢ ( 𝜑 → ( ( 𝜓 ↔ 𝜃 ) ↔ ( 𝜒 ↔ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	bibi2d	⊢ ( 𝜑 → ( ( 𝜃 ↔ 𝜓 ) ↔ ( 𝜃 ↔ 𝜒 ) ) )
3		bicom	⊢ ( ( 𝜓 ↔ 𝜃 ) ↔ ( 𝜃 ↔ 𝜓 ) )
4		bicom	⊢ ( ( 𝜒 ↔ 𝜃 ) ↔ ( 𝜃 ↔ 𝜒 ) )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( ( 𝜓 ↔ 𝜃 ) ↔ ( 𝜒 ↔ 𝜃 ) ) )