Theorem sbccomlem 2832

Description: Lemma for sbccom 2833. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Assertion

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

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem sbccomlem

Step	Hyp	Ref	Expression
1		excom 1554	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. y E. x ( x = A /\ ( y = B /\ ph ) ) )
2		exdistr 1787	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
3		an12 495	. . . . . . 7 |- ( ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ ( x = A /\ ph ) ) )
4	3	exbii 1496	. . . . . 6 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( y = B /\ ( x = A /\ ph ) ) )
5		19.42v 1786	. . . . . 6 |- ( E. x ( y = B /\ ( x = A /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
6	4, 5	bitri 173	. . . . 5 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
7	6	exbii 1496	. . . 4 |- ( E. y E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
8	1, 2, 7	3bitr3i 199	. . 3 |- ( E. x ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
9		sbc5 2787	. . 3 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
10		sbc5 2787	. . 3 |- ( [. B / y ]. E. x ( x = A /\ ph ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
11	8, 9, 10	3bitr4i 201	. 2 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> [. B / y ]. E. x ( x = A /\ ph ) )
12		sbc5 2787	. . 3 |- ( [. B / y ]. ph <-> E. y ( y = B /\ ph ) )
13	12	sbcbii 2818	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / x ]. E. y ( y = B /\ ph ) )
14		sbc5 2787	. . 3 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
15	14	sbcbii 2818	. 2 |- ( [. B / y ]. [. A / x ]. ph <-> [. B / y ]. E. x ( x = A /\ ph ) )
16	11, 13, 15	3bitr4i 201	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by:  sbccom  2833