Theorem nfcsb 3119

Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfcsb	⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Proof of Theorem nfcsb

Step	Hyp	Ref	Expression
1		nftru 1477	. . 3 ⊢ Ⅎ𝑦⊤
2		nfcsb.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfcsb.2	. . . 4 ⊢ Ⅎ𝑥𝐵
5	4	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐵)
6	1, 3, 5	nfcsbd 3117	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
7	6	mptru 1373	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1365  Ⅎwnfc 2323  ⦋csb 3081

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-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-sbc 2987  df-csb 3082

This theorem is referenced by:  cbvralcsf  3144  cbvrexcsf  3145  cbvreucsf  3146  cbvrabcsf  3147  elfvmptrab1  5653  fmptcof  5726  mpomptsx  6252  dmmpossx  6254  fmpox  6255  fmpoco  6271  dfmpo  6278  f1od2  6290  nfsum  11503  fsum2dlemstep  11580  fisumcom2  11584  nfcprod  11701  fsumcncntop  14746