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Metamath Proof Explorer

Theorem mteqand

Description: A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023)

Ref Expression
Hypotheses mteqand.1 ⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
mteqand.2 ⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐶 = 𝐷 )
Assertion mteqand ( 𝜑 → 𝐴 ≠ 𝐵 )

Proof

Step Hyp Ref Expression
1 mteqand.1 ⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
2 mteqand.2 ⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐶 = 𝐷 )
3 1 neneqd ⊢ ( 𝜑 → ¬ 𝐶 = 𝐷 )
4 3 2 mtand ⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )
5 4 neqned ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )