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Theorem cosseq 35541
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 5064 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑥𝑢𝐵𝑥))
2 breq 5064 . . . . 5 (𝐴 = 𝐵 → (𝑢𝐴𝑦𝑢𝐵𝑦))
31, 2anbi12d 630 . . . 4 (𝐴 = 𝐵 → ((𝑢𝐴𝑥𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥𝑢𝐵𝑦)))
43exbidv 1915 . . 3 (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)))
54opabbidv 5128 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)})
6 df-coss 35529 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥𝑢𝐴𝑦)}
7 df-coss 35529 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥𝑢𝐵𝑦)}
85, 6, 73eqtr4g 2885 1 (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wex 1773   class class class wbr 5062  {copab 5124  ccoss 35324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-br 5063  df-opab 5125  df-coss 35529
This theorem is referenced by:  cosseqi  35542  cosseqd  35543  elfunsALTV  35795
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