Theorem nfcsb 3108

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 1476	. . 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 3106	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
7	6	mptru 1372	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1364  Ⅎwnfc 2318  ⦋csb 3071

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-sbc 2977  df-csb 3072

This theorem is referenced by:  cbvralcsf  3133  cbvrexcsf  3134  cbvreucsf  3135  cbvrabcsf  3136  elfvmptrab1  5625  fmptcof  5698  mpomptsx  6215  dmmpossx  6217  fmpox  6218  fmpoco  6234  dfmpo  6241  f1od2  6253  nfsum  11382  fsum2dlemstep  11459  fisumcom2  11463  nfcprod  11580  fsumcncntop  14439