Metamath Proof Explorer


Theorem ltsub1dd

Description: Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses leidd.1 φ A
ltnegd.2 φ B
ltadd1d.3 φ C
ltadd1dd.4 φ A < B
Assertion ltsub1dd φ A C < B C

Proof

Step Hyp Ref Expression
1 leidd.1 φ A
2 ltnegd.2 φ B
3 ltadd1d.3 φ C
4 ltadd1dd.4 φ A < B
5 1 2 3 ltsub1d φ A < B A C < B C
6 4 5 mpbid φ A C < B C