Theorem nfin 3231

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

Hypotheses

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

Assertion

Ref	Expression
nfin	⊢ Ⅎx(A ∩ B)

Proof of Theorem nfin

Step	Hyp	Ref	Expression
1		df-in 3214	. 2 ⊢ (A ∩ B) = ∼ (A ⩃ B)
2		nfbool.1	. . . 4 ⊢ ℲxA
3		nfbool.2	. . . 4 ⊢ ℲxB
4	2, 3	nfnin 3229	. . 3 ⊢ Ⅎx(A ⩃ B)
5	4	nfcompl 3230	. 2 ⊢ Ⅎx ∼ (A ⩃ B)
6	1, 5	nfcxfr 2487	1 ⊢ Ⅎx(A ∩ B)

Syntax hints:  Ⅎwnfc 2477   ⩃ cnin 3205   ∼ ccompl 3206   ∩ cin 3209

This theorem was proved from 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 theorem 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  df-compl 3213  df-in 3214

This theorem is referenced by:  nfdif  3233  csbing  3463  nfres  4937