Theorem nfss 3235

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 3234	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
4		nfra1 2575	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵
5	3, 4	nfxfr 1523	1 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵

Syntax hints:  Ⅎwnf 1509   ∈ wcel 2205  Ⅎwnfc 2373  ∀wral 2522   ⊆ wss 3214

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-in 3220  df-ss 3227

This theorem is referenced by:  ssrexf  3304  nfpw  3690  ssiun2s  4040  triun  4226  ssopab2b  4400  nffrfor  4474  tfis  4710  nfrel  4840  nffun  5380  nff  5510  fvmptssdm  5767  ssoprab2b  6118  funimass4f  6332  nfsum1  12066  nfsum  12067  nfcprod1  12265  nfcprod  12266