Theorem nfnin 3229

Description: Hypothesis builder for anti-intersection. (Contributed by SF, 2-Jan-2018.)

Hypotheses

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

Assertion

Ref	Expression
nfnin	⊢ Ⅎx(A ⩃ B)

Proof of Theorem nfnin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nin 3212	. 2 ⊢ (A ⩃ B) = {y ∣ (y ∈ A ⊼ y ∈ B)}
2		nfnin.1	. . . . 5 ⊢ ℲxA
3	2	nfel2 2502	. . . 4 ⊢ Ⅎx y ∈ A
4		nfnin.2	. . . . 5 ⊢ ℲxB
5	4	nfel2 2502	. . . 4 ⊢ Ⅎx y ∈ B
6	3, 5	nfnan 1825	. . 3 ⊢ Ⅎx(y ∈ A ⊼ y ∈ B)
7	6	nfab 2494	. 2 ⊢ Ⅎx{y ∣ (y ∈ A ⊼ y ∈ B)}
8	1, 7	nfcxfr 2487	1 ⊢ Ⅎx(A ⩃ B)

This proof depends on syntax axioms:   ⊼ wnan 1287   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477   ⩃ cnin 3205

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-nin 3212

This theorem is used by:  nfcompl  3230  nfin  3231  nfun  3232