Theorem csbnest1g 3189

Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
csbnest1g	|- (A e. V -> [_A / x]_[_B / x]_C = [_[_A / x]_B / x]_C)

Proof of Theorem csbnest1g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcsb1v 3169	. . . 4 |- F/_x[_y / x]_C
2	1	ax-gen 1546	. . 3 |- A.y F/_x[_y / x]_C
3		csbnestgf 3185	. . 3 |- ((A e. V /\ A.y F/_x[_y / x]_C) -> [_A / x]_[_B / y]_[_y / x]_C = [_[_A / x]_B / y]_[_y / x]_C)
4	2, 3	mpan2 652	. 2 |- (A e. V -> [_A / x]_[_B / y]_[_y / x]_C = [_[_A / x]_B / y]_[_y / x]_C)
5		csbco 3146	. . 3 |- [_B / y]_[_y / x]_C = [_B / x]_C
6	5	csbeq2i 3163	. 2 |- [_A / x]_[_B / y]_[_y / x]_C = [_A / x]_[_B / x]_C
7		csbco 3146	. 2 |- [_[_A / x]_B / y]_[_y / x]_C = [_[_A / x]_B / x]_C
8	4, 6, 7	3eqtr3g 2408	1 |- (A e. V -> [_A / x]_[_B / x]_C = [_[_A / x]_B / x]_C)

Syntax hints:   -> wi 4  A.wal 1540   = wceq 1642   e. wcel 1710   F/_wnfc 2477  [_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-or 359  df-an 360  df-3an 936  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-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbnest1gOLD  3190  csbidmg  3191