Theorem leeq2d 45348

Description: Specialization of breq2d 5118 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 5134	1 ⊢ (𝜑 → 𝐴 ≤ 𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  ℝcr 11223   ≤ cle 11368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107

This theorem is used by: (None)