Theorem leeq2d 45158

Description: Specialization of breq2d 5114 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
leeq2d.1	⊢ (𝜑 → 𝐴 ≤ 𝐶)
leeq2d.2	⊢ (𝜑 → 𝐶 = 𝐷)
leeq2d.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
leeq2d.4	⊢ (𝜑 → 𝐶 ∈ ℝ)

Assertion

Ref	Expression
leeq2d	⊢ (𝜑 → 𝐴 ≤ 𝐷)

Proof of Theorem leeq2d

Step	Hyp	Ref	Expression
1		leeq2d.1	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
2		leeq2d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3	1, 2	breqtrd 5130	1 ⊢ (𝜑 → 𝐴 ≤ 𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ℝcr 11199   ≤ cle 11344

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103

This theorem is used by: (None)