Theorem nfss 3190

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 3189	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
4		nfra1 2538	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵
5	3, 4	nfxfr 1498	1 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵

Syntax hints:  Ⅎwnf 1484   ∈ wcel 2177  Ⅎwnfc 2336  ∀wral 2485   ⊆ wss 3170

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-in 3176  df-ss 3183

This theorem is referenced by:  ssrexf  3259  nfpw  3633  ssiun2s  3976  triun  4162  ssopab2b  4330  nffrfor  4402  tfis  4638  nfrel  4767  nffun  5302  nff  5431  fvmptssdm  5676  ssoprab2b  6014  nfsum1  11737  nfsum  11738  nfcprod1  11935  nfcprod  11936