Theorem nfuni 4378

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 4374	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfuni.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfv 1830	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
4	2, 3	nfrex 2990	. . 3 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧
5	4	nfab 2755	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
6	1, 5	nfcxfr 2749	1 ⊢ Ⅎ𝑥∪ 𝐴

Syntax hints:  {cab 2596  Ⅎwnfc 2738  ∃wrex 2897  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-uni 4373

This theorem is referenced by:  nfiota1  5770  nfwrecs  7296  nfsup  8240  ptunimpt  21208  disjabrex  28777  disjabrexf  28778  nfesum1  29429  nfesum2  29430  bnj1398  30356  bnj1446  30367  bnj1447  30368  bnj1448  30369  bnj1466  30375  bnj1467  30376  bnj1519  30387  bnj1520  30388  bnj1525  30391  bnj1523  30393  dfon2lem3  30934  bj-xnex  32245  mptsnunlem  32361  ptrest  32578  heibor1  32779  nfunidALT2  33274  nfunidALT  33275  disjinfi  38375  stoweidlem28  38921  stoweidlem59  38952  fourierdlem80  39079  smfresal  39673  smfpimbor1lem2  39684  nfsetrecs  42232