Theorem nfcsb1d 3547

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 3534	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfv 1843	. . 3 |- F/ y ph
3		nfcsb1d.1	. . . 4 |- ( ph -> F/_ x A )
4	3	nfsbc1d 3453	. . 3 |- ( ph -> F/ x [. A / x ]. y e. B )
5	2, 4	nfabd 2785	. 2 |- ( ph -> F/_ x { y | [. A / x ]. y e. B } )
6	1, 5	nfcxfrd 2763	1 |- ( ph -> F/_ x [_ A / x ]_ B )

Syntax hints:   -> wi 4   e. wcel 1990   {cab 2608   F/_wnfc 2751   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-sbc 3436  df-csb 3534

This theorem is referenced by:  nfcsb1  3548  riotaeqimp  6634