Theorem nfiunxy 3970

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

Hypotheses

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

Assertion

Ref	Expression
nfiunxy	⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiunxy

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3946	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiunxy.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiunxy.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2346	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrexw 2549	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2357	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2349	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2180  {cab 2195  Ⅎwnfc 2339  ∃wrex 2489  ∪ ciun 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-iun 3946

This theorem is referenced by:  iunab  3991