Theorem nfuni 4602

Description: Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
nfuni.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfuni	⊢ Ⅎ𝑥∪ 𝐴

Proof of Theorem nfuni

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfuni2 4598	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfuni.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 2009	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfrex 3153	. . 3 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2912	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2905	1 ⊢ Ⅎ𝑥∪ 𝐴

Syntax hints:  {cab 2751  Ⅎwnfc 2894  ∃wrex 3056  ∪ cuni 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-uni 4597

This theorem is referenced by:  nfiota1  6035  nfwrecs  7614  nfsup  8566  ptunimpt  21681  disjabrex  29846  disjabrexf  29847  nfesum1  30552  nfesum2  30553  bnj1398  31553  bnj1446  31564  bnj1447  31565  bnj1448  31566  bnj1466  31572  bnj1467  31573  bnj1519  31584  bnj1520  31585  bnj1525  31588  bnj1523  31590  dfon2lem3  32136  nffrecs  32225  mptsnunlem  33622  ptrest  33835  heibor1  34034  nfunidALT2  34928  nfunidALT  34929  disjinfi  40030  stoweidlem28  40885  stoweidlem59  40916  fourierdlem80  41043  smfresal  41638  smfpimbor1lem2  41649  nfafv2  41969  nfsetrecs  43105