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Metamath Proof Explorer

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 ) )