Theorem nfunid 4842

Description: Deduction version of nfuni 4843. (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 4838	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfv 1918	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1918	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfunid.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfvd 1919	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧)
6	3, 4, 5	nfrexd 3235	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
7	2, 6	nfabdw 2929	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧})
8	1, 7	nfcxfrd 2905	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4  {cab 2715  Ⅎwnfc 2886  ∃wrex 3064  ∪ cuni 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-uni 4837

This theorem is referenced by:  nfuni  4843  dfnfc2  4860  nfiotadw  6379  nfiotad  6381