Theorem nfcsb1d 3166

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

Hypothesis

Ref	Expression
nfcsb1d.1	|- (ph -> F/_xA)

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 3137	. 2 |- [_A / x]_B = {y | [.A / x ].y e. B}
2		nfv 1619	. . 3 |- F/yph
3		nfcsb1d.1	. . . 4 |- (ph -> F/_xA)
4	3	nfsbc1d 3063	. . 3 |- (ph -> F/x [.A / x ].y e. B)
5	2, 4	nfabd 2508	. 2 |- (ph -> F/_x{y | [.A / x ].y e. B})
6	1, 5	nfcxfrd 2487	1 |- (ph -> F/_x[_A / x]_B)

Syntax hints:   -> wi 4   e. wcel 1710  {cab 2339   F/_wnfc 2476   [.wsbc 3046  [_csb 3136

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 2478  df-sbc 3047  df-csb 3137

This theorem is referenced by:  nfcsb1  3167