Theorem sbccomlem 2913

Description: Lemma for sbccom 2914. (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 1599	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. y E. x ( x = A /\ ( y = B /\ ph ) ) )
2		exdistr 1835	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
3		an12 528	. . . . . . 7 |- ( ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ ( x = A /\ ph ) ) )
4	3	exbii 1541	. . . . . 6 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( y = B /\ ( x = A /\ ph ) ) )
5		19.42v 1834	. . . . . 6 |- ( E. x ( y = B /\ ( x = A /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
6	4, 5	bitri 182	. . . . 5 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
7	6	exbii 1541	. . . 4 |- ( E. y E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
8	1, 2, 7	3bitr3i 208	. . 3 |- ( E. x ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
9		sbc5 2863	. . 3 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
10		sbc5 2863	. . 3 |- ( [. B / y ]. E. x ( x = A /\ ph ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
11	8, 9, 10	3bitr4i 210	. 2 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> [. B / y ]. E. x ( x = A /\ ph ) )
12		sbc5 2863	. . 3 |- ( [. B / y ]. ph <-> E. y ( y = B /\ ph ) )
13	12	sbcbii 2898	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / x ]. E. y ( y = B /\ ph ) )
14		sbc5 2863	. . 3 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
15	14	sbcbii 2898	. 2 |- ( [. B / y ]. [. A / x ]. ph <-> [. B / y ]. E. x ( x = A /\ ph ) )
16	11, 13, 15	3bitr4i 210	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1289   E.wex 1426   [.wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841

This theorem is referenced by:  sbccom  2914