Theorem nfcsb1d 3038

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

Hypothesis

Ref	Expression
nfcsb1d.1	|- ( ph -> F/_ x A )

Assertion

Ref	Expression
nfcsb1d	|- ( ph -> F/_ x [_ A / x ]_ B )

Proof of Theorem nfcsb1d

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3008	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfv 1509	. . 3 |- F/ y ph
3		nfcsb1d.1	. . . 4 |- ( ph -> F/_ x A )
4	3	nfsbc1d 2929	. . 3 |- ( ph -> F/ x [. A / x ]. y e. B )
5	2, 4	nfabd 2301	. 2 |- ( ph -> F/_ x { y | [. A / x ]. y e. B } )
6	1, 5	nfcxfrd 2280	1 |- ( ph -> F/_ x [_ A / x ]_ B )

Syntax hints:   -> wi 4   e. wcel 1481   {cab 2126   F/_wnfc 2269   [.wsbc 2913   [_csb 3007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-sbc 2914  df-csb 3008

This theorem is referenced by:  nfcsb1  3039