Theorem nfinOLD 4174

Description: Obsolete version of nfin 4173 as of 14-May-2025. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfin.1	⊢ Ⅎ𝑥𝐴
nfin.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfinOLD	⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Proof of Theorem nfinOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfin5 3906	. 2 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
2		nfin.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	2	nfcri 2887	. . 3 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
4		nfin.1	. . 3 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfrabw 3433	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵}
6	1, 5	nfcxfr 2893	1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)

Syntax hints:   ∈ wcel 2113  Ⅎwnfc 2880  {crab 3396   ∩ cin 3897

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 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rab 3397  df-in 3905

This theorem is referenced by: (None)