Theorem cosseq 39416

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.

Step	Hyp	Ref	Expression
1		breq 5105	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑥 ↔ 𝑢𝐵𝑥))
2		breq 5105	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑦 ↔ 𝑢𝐵𝑦))
3	1, 2	anbi12d 644	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)))
4	3	exbidv 1954	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)))
5	4	opabbidv 5171	. 2 ⊢ (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)})
6		df-coss 39401	. 2 ⊢ ≀ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)}
7		df-coss 39401	. 2 ⊢ ≀ 𝐵 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)}
8	5, 6, 7	3eqtr4g 2821	1 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   class class class wbr 5103  {copab 5167   ≀ ccoss 39083

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-br 5104  df-opab 5168  df-coss 39401

This theorem is used by:  cosseqi  39417  cosseqd  39418  elfunsALTV  39677