Theorem nfiu1 3938

Description: Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)

Assertion

Ref	Expression
nfiu1	⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfiu1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 3910	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfre1 2533	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2337	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2329	1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2160  {cab 2175  Ⅎwnfc 2319  ∃wrex 2469  ∪ ciun 3908

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-iun 3910

This theorem is referenced by:  ssiun2s  3952  triun  4136  eliunxp  4791  opeliunxp2  4792  opeliunxp2f  6278  ixpf  6761  ctiunctlemfo  12570