Theorem csbcom2fi 33934

Description: Commutative law for double class substitution in a class, with non free variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)

Hypotheses

Ref	Expression
csbcom2fi.1	|- A e. _V
csbcom2fi.2	|- F/_ y A
csbcom2fi.3	|- [_ A / x ]_ B = C
csbcom2fi.4	|- [_ A / x ]_ D = E

Assertion

Ref	Expression
csbcom2fi	|- [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ E

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)   D( x, y)   E( x, y)

Proof of Theorem csbcom2fi

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3534	. . . . 5 |- [_ A / x ]_ [_ B / y ]_ D = { z | [. A / x ]. z e. [_ B / y ]_ D }
2	1	abeq2i 2735	. . . 4 |- ( z e. [_ A / x ]_ [_ B / y ]_ D <-> [. A / x ]. z e. [_ B / y ]_ D )
3		df-csb 3534	. . . . . 6 |- [_ B / y ]_ D = { z | [. B / y ]. z e. D }
4	3	abeq2i 2735	. . . . 5 |- ( z e. [_ B / y ]_ D <-> [. B / y ]. z e. D )
5	4	sbcbii 3491	. . . 4 |- ( [. A / x ]. z e. [_ B / y ]_ D <-> [. A / x ]. [. B / y ]. z e. D )
6	2, 5	bitri 264	. . 3 |- ( z e. [_ A / x ]_ [_ B / y ]_ D <-> [. A / x ]. [. B / y ]. z e. D )
7		csbcom2fi.1	. . . 4 |- A e. _V
8		csbcom2fi.2	. . . 4 |- F/_ y A
9		csbcom2fi.3	. . . 4 |- [_ A / x ]_ B = C
10		df-csb 3534	. . . . . 6 |- [_ A / x ]_ D = { z | [. A / x ]. z e. D }
11	10	abeq2i 2735	. . . . 5 |- ( z e. [_ A / x ]_ D <-> [. A / x ]. z e. D )
12		csbcom2fi.4	. . . . . 6 |- [_ A / x ]_ D = E
13	12	eleq2i 2693	. . . . 5 |- ( z e. [_ A / x ]_ D <-> z e. E )
14	11, 13	bitr3i 266	. . . 4 |- ( [. A / x ]. z e. D <-> z e. E )
15	7, 8, 9, 14	sbccom2fi 33932	. . 3 |- ( [. A / x ]. [. B / y ]. z e. D <-> [. C / y ]. z e. E )
16		sbcel2 3989	. . 3 |- ( [. C / y ]. z e. E <-> z e. [_ C / y ]_ E )
17	6, 15, 16	3bitri 286	. 2 |- ( z e. [_ A / x ]_ [_ B / y ]_ D <-> z e. [_ C / y ]_ E )
18	17	eqriv 2619	1 |- [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ E

Syntax hints:   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [.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-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)