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Metamath Proof Explorer

Theorem 3imtr3d

Description: More general version of 3imtr3i . Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996)

Ref Expression
Hypotheses 3imtr3d.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3imtr3d.2 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
3imtr3d.3 ⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
Assertion 3imtr3d ( 𝜑 → ( 𝜃 → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 3imtr3d.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 3imtr3d.2 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
3 3imtr3d.3 ⊢ ( 𝜑 → ( 𝜒 ↔ 𝜏 ) )
4 1 3 sylibd ⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )
5 2 4 sylbird ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )