Theorem nfcsbd 3170

Description: Deduction version of nfcsb 3171. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfcsbd.1	|- F/yph
nfcsbd.2	|- (ph -> F/_xA)
nfcsbd.3	|- (ph -> F/_xB)

Assertion

Ref	Expression
nfcsbd	|- (ph -> F/_x[_A / y]_B)

Proof of Theorem nfcsbd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3138	. 2 |- [_A / y]_B = {z | [.A / y ].z e. B}
2		nfv 1619	. . 3 |- F/zph
3		nfcsbd.1	. . . 4 |- F/yph
4		nfcsbd.2	. . . 4 |- (ph -> F/_xA)
5		nfcsbd.3	. . . . 5 |- (ph -> F/_xB)
6	5	nfcrd 2503	. . . 4 |- (ph -> F/x z e. B)
7	3, 4, 6	nfsbcd 3067	. . 3 |- (ph -> F/x [.A / y ].z e. B)
8	2, 7	nfabd 2509	. 2 |- (ph -> F/_x{z | [.A / y ].z e. B})
9	1, 8	nfcxfrd 2488	1 |- (ph -> F/_x[_A / y]_B)

Syntax hints:   -> wi 4   F/wnf 1544   e. wcel 1710  {cab 2339   F/_wnfc 2477   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048  df-csb 3138

This theorem is referenced by:  nfcsb  3171