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Theorem cosseqd 38410
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 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
cosseqd (𝜑 → ≀ 𝐴 = ≀ 𝐵)

Proof of Theorem cosseqd
StepHypRef Expression
1 cosseqd.1 . 2 (𝜑𝐴 = 𝐵)
2 cosseq 38408 . 2 (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
31, 2syl 17 1 (𝜑 → ≀ 𝐴 = ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-br 5149  df-opab 5211  df-coss 38393
This theorem is referenced by:  relbrcoss  38428  elcoeleqvrels  38577  releldmqscoss  38642  eldisjs  38704
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