Theorem nfiu1OLD 4981

Description: Obsolete version of nfiu1 4980 as of 14-May-2025. (Contributed by NM, 12-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nfiu1OLD	⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfiu1OLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4946	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfre1 3259	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2902	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2894	1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2113  {cab 2712  Ⅎwnfc 2881  ∃wrex 3058  ∪ ciun 4944

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-rex 3059  df-iun 4946

This theorem is referenced by: (None)