Theorem nfiung 4984

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2404. See nfiun 4982 for a version with more disjoint variable conditions, but not requiring ax-13 2404. (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 4952	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiung.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiung.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2917	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrex 3363	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfabg 2932	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2923	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2143  {cab 2741  Ⅎwnfc 2910  ∃wrex 3087  ∪ ciun 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-iun 4952

This theorem is referenced by: (None)