Theorem nfuni 3898

Description: Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
nfuni.1	⊢ ℲxA

Assertion

Ref	Expression
nfuni	⊢ Ⅎx∪A

Proof of Theorem nfuni

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

Step	Hyp	Ref	Expression
1		dfuni2 3894	. 2 ⊢ ∪A = {y ∣ ∃z ∈ A y ∈ z}
2		nfuni.1	. . . 4 ⊢ ℲxA
3		nfv 1619	. . . 4 ⊢ Ⅎx y ∈ z
4	2, 3	nfrex 2670	. . 3 ⊢ Ⅎx∃z ∈ A y ∈ z
5	4	nfab 2494	. 2 ⊢ Ⅎx{y ∣ ∃z ∈ A y ∈ z}
6	1, 5	nfcxfr 2487	1 ⊢ Ⅎx∪A

Syntax hints:   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  ∃wrex 2616  ∪cuni 3892

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-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 2479  df-ral 2620  df-rex 2621  df-uni 3893

This theorem is referenced by:  nfiota1  4342