Theorem cosseqd 38429

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 38427	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ≀ ccoss 38182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-br 5144  df-opab 5206  df-coss 38412

This theorem is referenced by:  relbrcoss  38447  elcoeleqvrels  38596  releldmqscoss  38661  eldisjs  38723