Theorem nfiun 3996

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiun.1	⊢ ℲyA
nfiun.2	⊢ ℲyB

Assertion

Ref	Expression
nfiun	⊢ Ⅎy∪x ∈ A B

Proof of Theorem nfiun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3972	. 2 ⊢ ∪x ∈ A B = {z ∣ ∃x ∈ A z ∈ B}
2		nfiun.1	. . . 4 ⊢ ℲyA
3		nfiun.2	. . . . 5 ⊢ ℲyB
4	3	nfcri 2484	. . . 4 ⊢ Ⅎy z ∈ B
5	2, 4	nfrex 2670	. . 3 ⊢ Ⅎy∃x ∈ A z ∈ B
6	5	nfab 2494	. 2 ⊢ Ⅎy{z ∣ ∃x ∈ A z ∈ B}
7	1, 6	nfcxfr 2487	1 ⊢ Ⅎy∪x ∈ A B

Syntax hints:   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  ∃wrex 2616  ∪ciun 3970

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 2479  df-ral 2620  df-rex 2621  df-iun 3972

This theorem is referenced by:  iunab  4013