Theorem leeq1d 44601

Description: Specialization of breq1d 5082 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 5096	1 ⊢ (𝜑 → 𝐵 ≤ 𝐶)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  ℝcr 11028   ≤ cle 11171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073

This theorem is referenced by: (None)