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Metamath Proof Explorer

Theorem pm2.21ddne

Description: A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses pm2.21ddne.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
pm2.21ddne.2 ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
Assertion pm2.21ddne ( 𝜑 → 𝜓 )

Proof

Step Hyp Ref Expression
1 pm2.21ddne.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 pm2.21ddne.2 ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
3 2 neneqd ⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )
4 1 3 pm2.21dd ⊢ ( 𝜑 → 𝜓 )