Theorem nfunid 3779

Description: Deduction version of nfuni 3778. (Contributed by NM, 18-Feb-2013.)

Hypothesis

Ref	Expression
nfunid.3	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfunid	⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Proof of Theorem nfunid

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

Step	Hyp	Ref	Expression
1		dfuni2 3774	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfv 1508	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1508	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfunid.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfvd 1509	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧)
6	3, 4, 5	nfrexdxy 2491	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
7	2, 6	nfabd 2319	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧})
8	1, 7	nfcxfrd 2297	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4  {cab 2143  Ⅎwnfc 2286  ∃wrex 2436  ∪ cuni 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-uni 3773

This theorem is referenced by:  dfnfc2  3790  nfiotadw  5137