Theorem dfsb2 2168

Description: An alternate definition of proper substitution that, like df-sb 1788, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)

Assertion

Ref	Expression
dfsb2	|- ( [ y / x ] ph <-> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )

Proof of Theorem dfsb2

Step	Hyp	Ref	Expression
1		sp 1911	. . . 4 |- ( A. x x = y -> x = y )
2		sbequ2 1789	. . . . 5 |- ( x = y -> ( [ y / x ] ph -> ph ) )
3	2	sps 1917	. . . 4 |- ( A. x x = y -> ( [ y / x ] ph -> ph ) )
4		orc 387	. . . 4 |- ( ( x = y /\ ph ) -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
5	1, 3, 4	syl6an 548	. . 3 |- ( A. x x = y -> ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) )
6		sb4 2151	. . . 4 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
7		olc 386	. . . 4 |- ( A. x ( x = y -> ph ) -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
8	6, 7	syl6 35	. . 3 |- ( -. A. x x = y -> ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) )
9	5, 8	pm2.61i 168	. 2 |- ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
10		sbequ1 2047	. . . 4 |- ( x = y -> ( ph -> [ y / x ] ph ) )
11	10	imp 431	. . 3 |- ( ( x = y /\ ph ) -> [ y / x ] ph )
12		sb2 2147	. . 3 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
13	11, 12	jaoi 381	. 2 |- ( ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) -> [ y / x ] ph )
14	9, 13	impbii 191	1 |- ( [ y / x ] ph <-> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   A.wal 1436   [wsb 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-12 1906  ax-13 2054

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1661  df-nf 1665  df-sb 1788

This theorem is referenced by:  dfsb3  2169