Theorem nfunidALT 39017

Description: Deduction version of nfuni 4863. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfunidALT.1	⊢ (𝜑 → Ⅎ𝑥𝐴)

Assertion

Ref	Expression
nfunidALT	⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Proof of Theorem nfunidALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfunidALT.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
2		abidnf 3656	. . 3 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴)
3	2	unieqd 4869	. 2 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴)
4		nfaba1 2902	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
5	4	nfuni 4863	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
6	1, 3, 5	nfded 39014	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2111  {cab 2709  Ⅎwnfc 2879  ∪ cuni 4856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3914  df-uni 4857

This theorem is referenced by: (None)