Theorem nfuni 4594

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 4590	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfuni.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1992	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfrex 3145	. . 3 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2907	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2900	1 ⊢ Ⅎ𝑥∪ 𝐴

Syntax hints:  {cab 2746  Ⅎwnfc 2889  ∃wrex 3051  ∪ cuni 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-uni 4589

This theorem is referenced by:  nfiota1  6014  nfwrecs  7578  nfsup  8522  ptunimpt  21600  disjabrex  29702  disjabrexf  29703  nfesum1  30411  nfesum2  30412  bnj1398  31409  bnj1446  31420  bnj1447  31421  bnj1448  31422  bnj1466  31428  bnj1467  31429  bnj1519  31440  bnj1520  31441  bnj1525  31444  bnj1523  31446  dfon2lem3  31995  nffrecs  32084  mptsnunlem  33496  ptrest  33721  heibor1  33922  nfunidALT2  34759  nfunidALT  34760  disjinfi  39879  stoweidlem28  40748  stoweidlem59  40779  fourierdlem80  40906  smfresal  41501  smfpimbor1lem2  41512  nfsetrecs  42943