Theorem sbccom2fi 33932

Description: Commutative law for double class substitution, with non free variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)

Hypotheses

Ref	Expression
sbccom2fi.1	|- A e. _V
sbccom2fi.2	|- F/_ y A
sbccom2fi.3	|- [_ A / x ]_ B = C
sbccom2fi.4	|- ( [. A / x ]. ph <-> ps )

Assertion

Ref	Expression
sbccom2fi	|- ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ps )

Distinct variable group:   x, y

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

Proof of Theorem sbccom2fi

Step	Hyp	Ref	Expression
1		sbccom2fi.1	. . 3 |- A e. _V
2		sbccom2fi.2	. . 3 |- F/_ y A
3	1, 2	sbccom2f 33931	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )
4		sbccom2fi.3	. . 3 |- [_ A / x ]_ B = C
5		dfsbcq 3437	. . 3 |- ( [_ A / x ]_ B = C -> ( [. [_ A / x ]_ B / y ]. [. A / x ]. ph <-> [. C / y ]. [. A / x ]. ph ) )
6	4, 5	ax-mp 5	. 2 |- ( [. [_ A / x ]_ B / y ]. [. A / x ]. ph <-> [. C / y ]. [. A / x ]. ph )
7		sbccom2fi.4	. . 3 |- ( [. A / x ]. ph <-> ps )
8	7	sbcbii 3491	. 2 |- ( [. C / y ]. [. A / x ]. ph <-> [. C / y ]. ps )
9	3, 6, 8	3bitri 286	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ps )

Syntax hints:   <-> wb 196   = 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-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

This theorem is referenced by:  csbcom2fi  33934