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Metamath Proof Explorer

Theorem zltlem1d

Description: Integer ordering relation, a deduction version. (Contributed by metakunt, 23-May-2024)

Ref Expression
Hypotheses zltlem1d.1 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
zltlem1d.2 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion zltlem1d ( 𝜑 → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )

Proof

Step Hyp Ref Expression
1 zltlem1d.1 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2 zltlem1d.2 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
3 zltlem1 ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )