Theorem sbccomlem 3508

Description: Lemma for sbccom 3509. (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 2042	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. y E. x ( x = A /\ ( y = B /\ ph ) ) )
2		exdistr 1919	. . . 4 |- ( E. x E. y ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
3		an12 838	. . . . . . 7 |- ( ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ ( x = A /\ ph ) ) )
4	3	exbii 1774	. . . . . 6 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. x ( y = B /\ ( x = A /\ ph ) ) )
5		19.42v 1918	. . . . . 6 |- ( E. x ( y = B /\ ( x = A /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
6	4, 5	bitri 264	. . . . 5 |- ( E. x ( x = A /\ ( y = B /\ ph ) ) <-> ( y = B /\ E. x ( x = A /\ ph ) ) )
7	6	exbii 1774	. . . 4 |- ( E. y E. x ( x = A /\ ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
8	1, 2, 7	3bitr3i 290	. . 3 |- ( E. x ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
9		sbc5 3460	. . 3 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
10		sbc5 3460	. . 3 |- ( [. B / y ]. E. x ( x = A /\ ph ) <-> E. y ( y = B /\ E. x ( x = A /\ ph ) ) )
11	8, 9, 10	3bitr4i 292	. 2 |- ( [. A / x ]. E. y ( y = B /\ ph ) <-> [. B / y ]. E. x ( x = A /\ ph ) )
12		sbc5 3460	. . 3 |- ( [. B / y ]. ph <-> E. y ( y = B /\ ph ) )
13	12	sbcbii 3491	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. A / x ]. E. y ( y = B /\ ph ) )
14		sbc5 3460	. . 3 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
15	14	sbcbii 3491	. 2 |- ( [. B / y ]. [. A / x ]. ph <-> [. B / y ]. E. x ( x = A /\ ph ) )
16	11, 13, 15	3bitr4i 292	1 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   [.wsbc 3435

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-v 3202  df-sbc 3436

This theorem is referenced by:  sbccom  3509