Theorem cosseqd 39559

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 39557	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	syl 18	1 ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ≀ ccoss 39224

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-br 5103  df-opab 5167  df-coss 39542

This theorem is used by:  relbrcoss  39577  elcoeleqvrels  39720  releldmqscoss  39786  eldisjs  39860