Theorem 2sb5nd 38776

Description: Equivalence for double substitution 2sb5 2443 without distinct x, y requirement. 2sb5nd 38776 is derived from 2sb5ndVD 39146. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
2sb5nd	|- ( ( -. A. x x = y \/ u = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )

Distinct variable groups:   x, u   y, u   x, v   y, v

Allowed substitution hints:   ph( x, y, v, u)

Proof of Theorem 2sb5nd

Step	Hyp	Ref	Expression
1		ax6e2ndeq 38775	. 2 |- ( ( -. A. x x = y \/ u = v ) <-> E. x E. y ( x = u /\ y = v ) )
2		anabs5 851	. . . 4 |- ( ( E. x E. y ( x = u /\ y = v ) /\ ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ) <-> ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
3		2pm13.193 38768	. . . . . . . . 9 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( ( x = u /\ y = v ) /\ ph ) )
4	3	exbii 1774	. . . . . . . 8 |- ( E. y ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> E. y ( ( x = u /\ y = v ) /\ ph ) )
5		nfs1v 2437	. . . . . . . . . 10 |- F/ y [ v / y ] ph
6	5	nfsb 2440	. . . . . . . . 9 |- F/ y [ u / x ] [ v / y ] ph
7	6	19.41 2103	. . . . . . . 8 |- ( E. y ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
8	4, 7	bitr3i 266	. . . . . . 7 |- ( E. y ( ( x = u /\ y = v ) /\ ph ) <-> ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
9	8	exbii 1774	. . . . . 6 |- ( E. x E. y ( ( x = u /\ y = v ) /\ ph ) <-> E. x ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
10		nfs1v 2437	. . . . . . 7 |- F/ x [ u / x ] [ v / y ] ph
11	10	19.41 2103	. . . . . 6 |- ( E. x ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
12	9, 11	bitr2i 265	. . . . 5 |- ( ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) )
13	12	anbi2i 730	. . . 4 |- ( ( E. x E. y ( x = u /\ y = v ) /\ ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ) <-> ( E. x E. y ( x = u /\ y = v ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )
14	2, 13	bitr3i 266	. . 3 |- ( ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. x E. y ( x = u /\ y = v ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )
15		pm5.32 668	. . 3 |- ( ( E. x E. y ( x = u /\ y = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) ) <-> ( ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. x E. y ( x = u /\ y = v ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) ) )
16	14, 15	mpbir 221	. 2 |- ( E. x E. y ( x = u /\ y = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )
17	1, 16	sylbi 207	1 |- ( ( -. A. x x = y \/ u = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   [wsb 1880

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-ne 2795  df-v 3202

This theorem is referenced by:  2uasbanh  38777  2uasbanhVD  39147