Theorem csbcom 3994

Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
csbcom	|- [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C

Distinct variable groups:   y, A   x, B   x, y

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

Proof of Theorem csbcom

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccom 3509	. . . 4 |- ( [. A / x ]. [. B / y ]. z e. C <-> [. B / y ]. [. A / x ]. z e. C )
2		sbcel2 3989	. . . . 5 |- ( [. B / y ]. z e. C <-> z e. [_ B / y ]_ C )
3	2	sbcbii 3491	. . . 4 |- ( [. A / x ]. [. B / y ]. z e. C <-> [. A / x ]. z e. [_ B / y ]_ C )
4		sbcel2 3989	. . . . 5 |- ( [. A / x ]. z e. C <-> z e. [_ A / x ]_ C )
5	4	sbcbii 3491	. . . 4 |- ( [. B / y ]. [. A / x ]. z e. C <-> [. B / y ]. z e. [_ A / x ]_ C )
6	1, 3, 5	3bitr3i 290	. . 3 |- ( [. A / x ]. z e. [_ B / y ]_ C <-> [. B / y ]. z e. [_ A / x ]_ C )
7		sbcel2 3989	. . 3 |- ( [. A / x ]. z e. [_ B / y ]_ C <-> z e. [_ A / x ]_ [_ B / y ]_ C )
8		sbcel2 3989	. . 3 |- ( [. B / y ]. z e. [_ A / x ]_ C <-> z e. [_ B / y ]_ [_ A / x ]_ C )
9	6, 7, 8	3bitr3i 290	. 2 |- ( z e. [_ A / x ]_ [_ B / y ]_ C <-> z e. [_ B / y ]_ [_ A / x ]_ C )
10	9	eqriv 2619	1 |- [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C

Syntax hints:   = wceq 1483   e. wcel 1990   [.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-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:  ovmpt2s  6784  fvmpt2curryd  7397