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Theorem cosseq 35551
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 5059 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑥𝑢𝐵𝑥))
2 breq 5059 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑦𝑢𝐵𝑦))
31, 2anbi12d 630 . . . 4 (𝐴 = 𝐵 → ((𝑢𝐴𝑥𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥𝑢𝐵𝑦)))
43exbidv 1913 . . 3 (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)))
54opabbidv 5123 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)})
6 df-coss 35539 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)}
7 df-coss 35539 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)}
85, 6, 73eqtr4g 2878 1 (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771   class class class wbr 5057  {copab 5119  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-br 5058  df-opab 5120  df-coss 35539
This theorem is referenced by:  cosseqi  35552  cosseqd  35553  elfunsALTV  35805
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