Theorem nfunid 4879

Description: Deduction version of nfuni 4880. (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 4875	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfv 1941	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1941	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfunid.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfvd 1942	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧)
6	3, 4, 5	nfrexdw 3317	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
7	2, 6	nfabdw 2952	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧})
8	1, 7	nfcxfrd 2930	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4  {cab 2747  Ⅎwnfc 2916  ∃wrex 3095  ∪ cuni 4873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-uni 4874

This theorem is referenced by:  nfuni  4880  dfnfc2  4895  nfiotadw  6493  nfiotad  6495