Theorem nfiu1OLD 5051

Description: Obsolete version of nfiu1 5050 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 5017	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfre1 3291	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2914	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2906	1 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2108  {cab 2717  Ⅎwnfc 2893  ∃wrex 3076  ∪ ciun 5015

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-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rex 3077  df-iun 5017

This theorem is referenced by: (None)