Theorem leeq1d 42908

Description: Specialization of breq1d 5159 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 5173	1 ⊢ (𝜑 → 𝐵 ≤ 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  ℝcr 11109   ≤ cle 11249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150

This theorem is referenced by:  imo72b2lem0  42917