Theorem unieq 3900

Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

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

Proof of Theorem unieq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 2808	. . 3 ⊢ (A = B → (∃x ∈ A y ∈ x ↔ ∃x ∈ B y ∈ x))
2	1	abbidv 2467	. 2 ⊢ (A = B → {y ∣ ∃x ∈ A y ∈ x} = {y ∣ ∃x ∈ B y ∈ x})
3		dfuni2 3893	. 2 ⊢ ∪A = {y ∣ ∃x ∈ A y ∈ x}
4		dfuni2 3893	. 2 ⊢ ∪B = {y ∣ ∃x ∈ B y ∈ x}
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ∪A = ∪B)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  ∪cuni 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-uni 3892

This theorem is referenced by:  unieqi  3901  unieqd  3902  uniintsn  3963  iununi  4050  pw1equn  4331  pw1eqadj  4332  nnadjoin  4520  pw1fnval  5851  pw1fnf1o  5855  brtcfn  6246