Theorem nfunid 4872

Description: Deduction version of nfuni 4873. (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 4868	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfv 1946	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1946	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfunid.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfvd 1947	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧)
6	3, 4, 5	nfrexdw 3308	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
7	2, 6	nfabdw 2943	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧})
8	1, 7	nfcxfrd 2921	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

This proof depends on syntax axioms:   → wi 4  {cab 2738  Ⅎwnfc 2907  ∃wrex 3086  ∪ cuni 4866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-uni 4867

This theorem is used by:  nfuni  4873  dfnfc2  4888  nfiotadw  6499  nfiotad  6501