Theorem nfiu1OLD 4970

Description: Obsolete version of nfiu1 4969 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 4935	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfre1 3262	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2904	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2896	1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2114  {cab 2714  Ⅎwnfc 2883  ∃wrex 3061  ∪ ciun 4933

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-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rex 3062  df-iun 4935

This theorem is referenced by: (None)