Theorem nfcsb1d 3080

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 3050	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfv 1521	. . 3 |- F/ y ph
3		nfcsb1d.1	. . . 4 |- ( ph -> F/_ x A )
4	3	nfsbc1d 2971	. . 3 |- ( ph -> F/ x [. A / x ]. y e. B )
5	2, 4	nfabd 2332	. 2 |- ( ph -> F/_ x { y | [. A / x ]. y e. B } )
6	1, 5	nfcxfrd 2310	1 |- ( ph -> F/_ x [_ A / x ]_ B )

Syntax hints:   -> wi 4   e. wcel 2141   {cab 2156   F/_wnfc 2299   [.wsbc 2955   [_csb 3049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956  df-csb 3050

This theorem is referenced by:  nfcsb1  3081