Metamath Proof Explorer


Theorem imbi12d

Description: Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993)

Ref Expression
Hypotheses imbi12d.1 ( 𝜑 → ( 𝜓𝜒 ) )
imbi12d.2 ( 𝜑 → ( 𝜃𝜏 ) )
Assertion imbi12d ( 𝜑 → ( ( 𝜓𝜃 ) ↔ ( 𝜒𝜏 ) ) )

Proof

Step Hyp Ref Expression
1 imbi12d.1 ( 𝜑 → ( 𝜓𝜒 ) )
2 imbi12d.2 ( 𝜑 → ( 𝜃𝜏 ) )
3 1 imbi1d ( 𝜑 → ( ( 𝜓𝜃 ) ↔ ( 𝜒𝜃 ) ) )
4 2 imbi2d ( 𝜑 → ( ( 𝜒𝜃 ) ↔ ( 𝜒𝜏 ) ) )
5 3 4 bitrd ( 𝜑 → ( ( 𝜓𝜃 ) ↔ ( 𝜒𝜏 ) ) )