Theorem unieq 1927

Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18.

Assertion

Ref	Expression
unieq	⊢ (A = B → ∪A = ∪B)

Proof of Theorem unieq

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . . 5 ⊢ (A = B → (y ∈ A ↔ y ∈ B))
2	1	anbi2d 468	. . . 4 ⊢ (A = B → ((x ∈ y ∧ y ∈ A) ↔ (x ∈ y ∧ y ∈ B)))
3	2	biexdv 936	. . 3 ⊢ (A = B → (∃y(x ∈ y ∧ y ∈ A) ↔ ∃y(x ∈ y ∧ y ∈ B)))
4	3	biabdv 1183	. 2 ⊢ (A = B → {x∣∃y(x ∈ y ∧ y ∈ A)} = {x∣∃y(x ∈ y ∧ y ∈ B)})
5		df-uni 1920	. 2 ⊢ ∪A = {x∣∃y(x ∈ y ∧ y ∈ A)}
6		df-uni 1920	. 2 ⊢ ∪B = {x∣∃y(x ∈ y ∧ y ∈ B)}
7	4, 5, 6	3eqtr4g 1147	1 ⊢ (A = B → ∪A = ∪B)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∪cuni 1919

This theorem is referenced by:  unieqi 1928  unieqd 1929  uniex 1947  uniexg 1948  euuni 1954  reucl 1957  treq 2047  limeq 2211  ordunisuc 2339  onuninsuc 2356  limsuclem 2360  unizlim 2364  orduninsuc 2365  fvex 2838  tz7.44-2 2967  rdglem2 2976  inf5 3472  trcl 3489  hsupval2t 5301

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-uni 1920

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