Theorem nfiu1OLD 5033

Description: Obsolete version of nfiu1 5032 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 4998	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfre1 3283	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2909	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2901	1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2106  {cab 2712  Ⅎwnfc 2888  ∃wrex 3068  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069  df-iun 4998

This theorem is referenced by: (None)