Theorem sbel2x 2459

Description: Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)

Assertion

Ref	Expression
sbel2x	|- ( ph <-> E. x E. y ( ( x = z /\ y = w ) /\ [ y / w ] [ x / z ] ph ) )

Distinct variable group:   x, y, ph

Allowed substitution hints:   ph( z, w)

Proof of Theorem sbel2x

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ y ph
2		nfv 1843	. . 3 |- F/ x ph
3	1, 2	2sb5rf 2451	. 2 |- ( ph <-> E. y E. x ( ( y = w /\ x = z ) /\ [ y / w ] [ x / z ] ph ) )
4		ancom 466	. . . 4 |- ( ( y = w /\ x = z ) <-> ( x = z /\ y = w ) )
5	4	anbi1i 731	. . 3 |- ( ( ( y = w /\ x = z ) /\ [ y / w ] [ x / z ] ph ) <-> ( ( x = z /\ y = w ) /\ [ y / w ] [ x / z ] ph ) )
6	5	2exbii 1775	. 2 |- ( E. y E. x ( ( y = w /\ x = z ) /\ [ y / w ] [ x / z ] ph ) <-> E. y E. x ( ( x = z /\ y = w ) /\ [ y / w ] [ x / z ] ph ) )
7		excom 2042	. 2 |- ( E. y E. x ( ( x = z /\ y = w ) /\ [ y / w ] [ x / z ] ph ) <-> E. x E. y ( ( x = z /\ y = w ) /\ [ y / w ] [ x / z ] ph ) )
8	3, 6, 7	3bitri 286	1 |- ( ph <-> E. x E. y ( ( x = z /\ y = w ) /\ [ y / w ] [ x / z ] ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

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

This theorem is referenced by: (None)