Theorem nfiinxy 3992

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiunxy.1	⊢ Ⅎ𝑦𝐴
nfiunxy.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfiinxy	⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiinxy

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3968	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiunxy.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiunxy.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2366	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfralxy 2568	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2377	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2369	1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2200  {cab 2215  Ⅎwnfc 2359  ∀wral 2508  ∩ ciin 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-iin 3968

This theorem is referenced by:  iinab  4027