Theorem nfii1 4487

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
nfii1	⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfii1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 4458	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		nfra1 2925	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	2	nfab 2755	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
4	1, 3	nfcxfr 2749	1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  ∀wral 2896  ∩ ciin 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-iin 4458

This theorem is referenced by:  dmiin  5290  scott0  8632  gruiin  9511  iooiinicc  38616  iooiinioc  38630  fnlimfvre  38741  fnlimabslt  38746  meaiininclem  39376  hspdifhsp  39506  smflimlem2  39658  smflim  39663