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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)

Colors of variables: wff setvar class

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

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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

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