Theorem 2pm13.193 38768

Description: pm13.193 38612 for two variables. pm13.193 38612 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 39139. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
2pm13.193	|- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( ( x = u /\ y = v ) /\ ph ) )

Proof of Theorem 2pm13.193

Step	Hyp	Ref	Expression
1		simpll 790	. . 3 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> x = u )
2		simplr 792	. . 3 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> y = v )
3		simpr 477	. . . . 5 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> [ u / x ] [ v / y ] ph )
4		sbequ2 1882	. . . . 5 |- ( x = u -> ( [ u / x ] [ v / y ] ph -> [ v / y ] ph ) )
5	1, 3, 4	sylc 65	. . . 4 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> [ v / y ] ph )
6		sbequ2 1882	. . . 4 |- ( y = v -> ( [ v / y ] ph -> ph ) )
7	2, 5, 6	sylc 65	. . 3 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> ph )
8	1, 2, 7	jca31 557	. 2 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) -> ( ( x = u /\ y = v ) /\ ph ) )
9		simpll 790	. . 3 |- ( ( ( x = u /\ y = v ) /\ ph ) -> x = u )
10		simplr 792	. . 3 |- ( ( ( x = u /\ y = v ) /\ ph ) -> y = v )
11		simpr 477	. . . . 5 |- ( ( ( x = u /\ y = v ) /\ ph ) -> ph )
12		sbequ1 2110	. . . . 5 |- ( y = v -> ( ph -> [ v / y ] ph ) )
13	10, 11, 12	sylc 65	. . . 4 |- ( ( ( x = u /\ y = v ) /\ ph ) -> [ v / y ] ph )
14		sbequ1 2110	. . . 4 |- ( x = u -> ( [ v / y ] ph -> [ u / x ] [ v / y ] ph ) )
15	9, 13, 14	sylc 65	. . 3 |- ( ( ( x = u /\ y = v ) /\ ph ) -> [ u / x ] [ v / y ] ph )
16	9, 10, 15	jca31 557	. 2 |- ( ( ( x = u /\ y = v ) /\ ph ) -> ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
17	8, 16	impbii 199	1 |- ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( ( x = u /\ y = v ) /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   [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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881

This theorem is referenced by:  2sb5nd  38776  2sb5ndVD  39146  2sb5ndALT  39168