Theorem nfcsb1d 3132

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 3102	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfv 1552	. . 3 |- F/ y ph
3		nfcsb1d.1	. . . 4 |- ( ph -> F/_ x A )
4	3	nfsbc1d 3022	. . 3 |- ( ph -> F/ x [. A / x ]. y e. B )
5	2, 4	nfabd 2370	. 2 |- ( ph -> F/_ x { y | [. A / x ]. y e. B } )
6	1, 5	nfcxfrd 2348	1 |- ( ph -> F/_ x [_ A / x ]_ B )

Syntax hints:   -> wi 4   e. wcel 2178   {cab 2193   F/_wnfc 2337   [.wsbc 3005   [_csb 3101

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-sbc 3006  df-csb 3102

This theorem is referenced by:  nfcsb1  3133