Theorem nfunid 4873

Description: Deduction version of nfuni 4874. (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 4869	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}
2		nfv 1947	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfv 1947	. . . 4 ⊢ Ⅎ𝑧𝜑
4		nfunid.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfvd 1948	. . . 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 4867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-uni 4868

This theorem is used by:  nfuni  4874  dfnfc2  4889  nfiotadw  6492  nfiotad  6494