Theorem nfss 3266

Description: If x is not free in A and B, it is not free in A ⊆ B. (Contributed by NM, 27-Dec-1996.)

Hypotheses

Ref	Expression
dfss2f.1	⊢ ℲxA
dfss2f.2	⊢ ℲxB

Assertion

Ref	Expression
nfss	⊢ Ⅎx A ⊆ B

Proof of Theorem nfss

Step	Hyp	Ref	Expression
1		dfss2f.1	. . 3 ⊢ ℲxA
2		dfss2f.2	. . 3 ⊢ ℲxB
3	1, 2	dfss3f 3265	. 2 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)
4		nfra1 2664	. 2 ⊢ Ⅎx∀x ∈ A x ∈ B
5	3, 4	nfxfr 1570	1 ⊢ Ⅎx A ⊆ B

Syntax hints:  Ⅎwnf 1544   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614   ⊆ wss 3257

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 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  nfpw  3733  ssiun2s  4010  ssopab2b  4713  nffun  5130  nff  5221  fvmptss  5705