Theorem cosseqd 37762

Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵, deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.)

Hypothesis

Ref	Expression
cosseqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cosseqd	⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵)

Proof of Theorem cosseqd

Step	Hyp	Ref	Expression
1		cosseqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		cosseq 37760	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ≀ ccoss 37507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-br 5149  df-opab 5211  df-coss 37745

This theorem is referenced by:  relbrcoss  37780  elcoeleqvrels  37929  releldmqscoss  37994  eldisjs  38056