Theorem csbnestgf 3185

Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Assertion

Ref	Expression
csbnestgf	|- ((A e. V /\ A.y F/_xC) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)

Proof of Theorem csbnestgf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. . 3 |- (A e. V -> A e. _V)
2		df-csb 3138	. . . . . . 7 |- [_B / y]_C = {z | [.B / y ].z e. C}
3	2	abeq2i 2461	. . . . . 6 |- (z e. [_B / y]_C <-> [.B / y ].z e. C)
4	3	sbcbii 3102	. . . . 5 |- ( [.A / x ].z e. [_B / y]_C <-> [.A / x ]. [.B / y ].z e. C)
5		nfcr 2482	. . . . . . 7 |- ( F/_xC -> F/x z e. C)
6	5	alimi 1559	. . . . . 6 |- (A.y F/_xC -> A.y F/x z e. C)
7		sbcnestgf 3184	. . . . . 6 |- ((A e. _V /\ A.y F/x z e. C) -> ( [.A / x ]. [.B / y ].z e. C <-> [.[_A / x]_B / y ].z e. C))
8	6, 7	sylan2 460	. . . . 5 |- ((A e. _V /\ A.y F/_xC) -> ( [.A / x ]. [.B / y ].z e. C <-> [.[_A / x]_B / y ].z e. C))
9	4, 8	syl5bb 248	. . . 4 |- ((A e. _V /\ A.y F/_xC) -> ( [.A / x ].z e. [_B / y]_C <-> [.[_A / x]_B / y ].z e. C))
10	9	abbidv 2468	. . 3 |- ((A e. _V /\ A.y F/_xC) -> {z | [.A / x ].z e. [_B / y]_C} = {z | [.[_A / x]_B / y ].z e. C})
11	1, 10	sylan 457	. 2 |- ((A e. V /\ A.y F/_xC) -> {z | [.A / x ].z e. [_B / y]_C} = {z | [.[_A / x]_B / y ].z e. C})
12		df-csb 3138	. 2 |- [_A / x]_[_B / y]_C = {z | [.A / x ].z e. [_B / y]_C}
13		df-csb 3138	. 2 |- [_[_A / x]_B / y]_C = {z | [.[_A / x]_B / y ].z e. C}
14	11, 12, 13	3eqtr4g 2410	1 |- ((A e. V /\ A.y F/_xC) -> [_A / x]_[_B / y]_C = [_[_A / x]_B / y]_C)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2477  _Vcvv 2860   [.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-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:  csbnestg  3187  csbnest1g  3189