Theorem nfunidALT2 36983

Description: Deduction version of nfuni 4846. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfunidALT2.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfunidALT2	⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Proof of Theorem nfunidALT2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2915	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
2	1	nfuni 4846	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
3		nfunidALT2.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfnfc1 2910	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
5		abidnf 3638	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴)
6	5	unieqd 4853	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴)
7	4, 6	nfceqdf 2902	. . 3 ⊢ (Ⅎ𝑥𝐴 → (Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} ↔ Ⅎ𝑥∪ 𝐴))
8	3, 7	syl 17	. 2 ⊢ (𝜑 → (Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} ↔ Ⅎ𝑥∪ 𝐴))
9	2, 8	mpbii 232	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840

This theorem is referenced by: (None)