Theorem nfcsb 3178

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 1515	. . 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 3176	. 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)
7	6	mptru 1407	1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵

Syntax hints:  ⊤wtru 1399  Ⅎwnfc 2373  ⦋csb 3140

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-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-sbc 3045  df-csb 3141

This theorem is referenced by:  cbvralcsf  3203  cbvrexcsf  3204  cbvreucsf  3205  cbvrabcsf  3206  elfvmptrab1  5774  fmptcof  5846  mpomptsx  6395  dmmpossx  6397  fmpox  6398  fmpoco  6414  dfmpo  6421  f1od2  6433  nfsum  12046  fsum2dlemstep  12124  fisumcom2  12128  nfcprod  12245  fsumcncntop  15449