Theorem sbccom2lem 33929

Description: Lemma for sbccom2 33930. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypothesis

Ref	Expression
sbccom2lem.1	|- A e. _V

Assertion

Ref	Expression
sbccom2lem	|- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )

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

Allowed substitution hints:   ph( x, y)   B( x)

Proof of Theorem sbccom2lem

Step	Hyp	Ref	Expression
1		sbcan 3478	. . . 4 |- ( [. A / x ]. ( y = B /\ ph ) <-> ( [. A / x ]. y = B /\ [. A / x ]. ph ) )
2		sbc5 3460	. . . 4 |- ( [. A / x ]. ( y = B /\ ph ) <-> E. x ( x = A /\ ( y = B /\ ph ) ) )
3		sbccom2lem.1	. . . . . 6 |- A e. _V
4	3	csbconstgi 33922	. . . . . 6 |- [_ A / x ]_ y = y
5		eqid 2622	. . . . . 6 |- [_ A / x ]_ B = [_ A / x ]_ B
6	3, 4, 5	sbceqi 33913	. . . . 5 |- ( [. A / x ]. y = B <-> y = [_ A / x ]_ B )
7	6	anbi1i 731	. . . 4 |- ( ( [. A / x ]. y = B /\ [. A / x ]. ph ) <-> ( y = [_ A / x ]_ B /\ [. A / x ]. ph ) )
8	1, 2, 7	3bitr3i 290	. . 3 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> ( y = [_ A / x ]_ B /\ [. A / x ]. ph ) )
9	8	exbii 1774	. 2 |- ( E. y E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. y ( y = [_ A / x ]_ B /\ [. A / x ]. ph ) )
10		sbc5 3460	. . . . 5 |- ( [. B / y ]. ph <-> E. y ( y = B /\ ph ) )
11	10	sbcbii 3491	. . . 4 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / x ]. E. y ( y = B /\ ph ) )
12		sbc5 3460	. . . 4 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
13	11, 12	bitri 264	. . 3 |- ( [. A / x ]. [. B / y ]. ph <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
14		19.42v 1918	. . . . . 6 |- ( E. y ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y ( y = B /\ ph ) ) )
15	14	bicomi 214	. . . . 5 |- ( ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. y ( x = A /\ ( y = B /\ ph ) ) )
16	15	exbii 1774	. . . 4 |- ( E. x ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. x E. y ( x = A /\ ( y = B /\ ph ) ) )
17		excom 2042	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. y E. x ( x = A /\ ( y = B /\ ph ) ) )
18	16, 17	bitri 264	. . 3 |- ( E. x ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. y E. x ( x = A /\ ( y = B /\ ph ) ) )
19	13, 18	bitri 264	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> E. y E. x ( x = A /\ ( y = B /\ ph ) ) )
20		sbc5 3460	. 2 |- ( [. [_ A / x ]_ B / y ]. [. A / x ]. ph <-> E. y ( y = [_ A / x ]_ B /\ [. A / x ]. ph ) )
21	9, 19, 20	3bitr4i 292	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _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-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:  sbccom2  33930