Theorem nfunidALT 39764

Description: Deduction version of nfuni 4879. (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 3665	. . 3 ⊢ (Ⅎ𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = 𝐴)
3	2	unieqd 4885	. 2 ⊢ (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} = ∪ 𝐴)
4		nfaba1 2933	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
5	4	nfuni 4879	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
6	1, 3, 5	nfded 39761	1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)

Syntax hints:   → wi 4  ∀wal 1568   ∈ wcel 2143  {cab 2741  Ⅎwnfc 2910  ∪ cuni 4872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-v 3457  df-ss 3922  df-uni 4873

This theorem is referenced by: (None)