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Theorem cosseq 36204
Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
Assertion
Ref Expression
cosseq (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

Proof of Theorem cosseq
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 5042 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑥𝑢𝐵𝑥))
2 breq 5042 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑦𝑢𝐵𝑦))
31, 2anbi12d 634 . . . 4 (𝐴 = 𝐵 → ((𝑢𝐴𝑥𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥𝑢𝐵𝑦)))
43exbidv 1928 . . 3 (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)))
54opabbidv 5106 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)})
6 df-coss 36192 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)}
7 df-coss 36192 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)}
85, 6, 73eqtr4g 2799 1 (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wex 1786   class class class wbr 5040  {copab 5102  ccoss 35988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-br 5041  df-opab 5103  df-coss 36192
This theorem is referenced by:  cosseqi  36205  cosseqd  36206  elfunsALTV  36458
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