Theorem nfunidALT 39416

Description: Deduction version of nfuni 4857. (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 3648	. . 3 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴)
3	2	unieqd 4863	. 2 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴)
4		nfaba1 2906	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
5	4	nfuni 4857	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
6	1, 3, 5	nfded 39413	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 2114  {cab 2714  Ⅎwnfc 2883  ∪ cuni 4850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-v 3431  df-ss 3906  df-uni 4851

This theorem is referenced by: (None)