Theorem leeq1d 38455

Description: Specialization of breq1d 4663 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 4677	1 ⊢ (𝜑 → 𝐵 ≤ 𝐶)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   class class class wbr 4653  ℝcr 9935   ≤ cle 10075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  imo72b2lem0  38465