Theorem nfunidALT2 38969

Description: Deduction version of nfuni 4881. (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 2900	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
2	1	nfuni 4881	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
3		nfunidALT2.1	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfnfc1 2895	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
5		abidnf 3676	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴)
6	5	unieqd 4887	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴)
7	4, 6	nfceqdf 2888	. . 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 233	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109  {cab 2708  Ⅎwnfc 2877  ∪ cuni 4874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-v 3452  df-ss 3934  df-uni 4875

This theorem is referenced by: (None)