Theorem cosseqi 39116

Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.)

Hypothesis

Ref	Expression
cosseqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
cosseqi	⊢ ≀ 𝐴 = ≀ 𝐵

Proof of Theorem cosseqi

Step	Hyp	Ref	Expression
1		cosseqi.1	. 2 ⊢ 𝐴 = 𝐵
2		cosseq 39115	. 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ≀ 𝐴 = ≀ 𝐵

Syntax hints:   = wceq 1568   ≀ ccoss 38782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-br 5115  df-opab 5179  df-coss 39100

This theorem is referenced by:  br1cossinres  39136  br1cossxrnres  39137  cosscnvid  39170