Theorem cosseqd 36551

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 36549	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ≀ ccoss 36333

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-br 5075  df-opab 5137  df-coss 36537

This theorem is referenced by:  relbrcoss  36564  elcoeleqvrels  36708  releldmqscoss  36772  eldisjs  36833