Theorem sbccomlem 3106

Description: Lemma for sbccom 3107. (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 1712	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. y E. x ( x = A /\ ( y = B /\ ph ) ) )
2		exdistr 1958	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
3		an12 563	. . . . . . 7 |- ( ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ ( x = A /\ ph ) ) )
4	3	exbii 1653	. . . . . 6 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( y = B /\ ( x = A /\ ph ) ) )
5		19.42v 1955	. . . . . 6 |- ( E. x ( y = B /\ ( x = A /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
6	4, 5	bitri 184	. . . . 5 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
7	6	exbii 1653	. . . 4 |- ( E. y E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
8	1, 2, 7	3bitr3i 210	. . 3 |- ( E. x ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
9		sbc5 3055	. . 3 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
10		sbc5 3055	. . 3 |- ( [. B / y ]. E. x ( x = A /\ ph ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
11	8, 9, 10	3bitr4i 212	. 2 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> [. B / y ]. E. x ( x = A /\ ph ) )
12		sbc5 3055	. . 3 |- ( [. B / y ]. ph <-> E. y ( y = B /\ ph ) )
13	12	sbcbii 3091	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / x ]. E. y ( y = B /\ ph ) )
14		sbc5 3055	. . 3 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
15	14	sbcbii 3091	. 2 |- ( [. B / y ]. [. A / x ]. ph <-> [. B / y ]. E. x ( x = A /\ ph ) )
16	11, 13, 15	3bitr4i 212	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   [.wsbc 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032

This theorem is referenced by:  sbccom  3107