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Metamath Proof Explorer

Theorem pm5.21nii

Description: Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999)

Ref Expression
Hypotheses pm5.21ni.1 ⊢ ( 𝜑 → 𝜓 )
pm5.21ni.2 ⊢ ( 𝜒 → 𝜓 )
pm5.21nii.3 ⊢ ( 𝜓 → ( 𝜑 ↔ 𝜒 ) )
Assertion pm5.21nii ( 𝜑 ↔ 𝜒 )

Proof

Step Hyp Ref Expression
1 pm5.21ni.1 ⊢ ( 𝜑 → 𝜓 )
2 pm5.21ni.2 ⊢ ( 𝜒 → 𝜓 )
3 pm5.21nii.3 ⊢ ( 𝜓 → ( 𝜑 ↔ 𝜒 ) )
4 1 2 pm5.21ni ⊢ ( ¬ 𝜓 → ( 𝜑 ↔ 𝜒 ) )
5 3 4 pm2.61i ⊢ ( 𝜑 ↔ 𝜒 )