Theorem nfunidALT 38926

Description: Deduction version of nfuni 4938. (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 3724	. . 3 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴)
3	2	unieqd 4944	. 2 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴)
4		nfaba1 2916	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
5	4	nfuni 4938	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
6	1, 3, 5	nfded 38923	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2108  {cab 2717  Ⅎwnfc 2893  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-ss 3993  df-uni 4932

This theorem is referenced by: (None)