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 ) ) )