Theorem nfiung 5033

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2366. See nfiun 5031 for a version with more disjoint variable conditions, but not requiring ax-13 2366. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfiung	⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfiung

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 5003	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiung.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiung.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2883	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrex 3359	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfabg 2899	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2890	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2099  {cab 2703  Ⅎwnfc 2876  ∃wrex 3060  ∪ ciun 5001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-iun 5003

This theorem is referenced by: (None)