Theorem nfss 3172

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 3171	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
4		nfra1 2525	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵
5	3, 4	nfxfr 1485	1 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵

Syntax hints:  Ⅎwnf 1471   ∈ wcel 2164  Ⅎwnfc 2323  ∀wral 2472   ⊆ wss 3153

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-in 3159  df-ss 3166

This theorem is referenced by:  ssrexf  3241  nfpw  3614  ssiun2s  3956  triun  4140  ssopab2b  4307  nffrfor  4379  tfis  4615  nfrel  4744  nffun  5277  nff  5400  fvmptssdm  5642  ssoprab2b  5975  nfsum1  11499  nfsum  11500  nfcprod1  11697  nfcprod  11698