Theorem leeq2d 44832

Description: Specialization of breq2d 5120 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 5136	1 ⊢ (𝜑 → 𝐴 ≤ 𝐷)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141   class class class wbr 5108  ℝcr 11098   ≤ cle 11243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109

This theorem is referenced by: (None)