Theorem leeq1d 44733

Description: Specialization of breq1d 5110 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
leeq1d.1	⊢ (𝜑 → 𝐴 ≤ 𝐶)
leeq1d.2	⊢ (𝜑 → 𝐴 = 𝐵)
leeq1d.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
leeq1d.4	⊢ (𝜑 → 𝐶 ∈ ℝ)

Assertion

Ref	Expression
leeq1d	⊢ (𝜑 → 𝐵 ≤ 𝐶)

Proof of Theorem leeq1d

Step	Hyp	Ref	Expression
1		leeq1d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		leeq1d.1	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
3	1, 2	eqbrtrrd 5124	1 ⊢ (𝜑 → 𝐵 ≤ 𝐶)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142   class class class wbr 5100  ℝcr 11072   ≤ cle 11217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101

This theorem is referenced by: (None)