Theorem dfsb2 2373

Description: An alternate definition of proper substitution that, like df-sb 1881, 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 2053	. . . 4 |- ( A. x x = y -> x = y )
2		sbequ2 1882	. . . . 5 |- ( x = y -> ( [ y / x ] ph -> ph ) )
3	2	sps 2055	. . . 4 |- ( A. x x = y -> ( [ y / x ] ph -> ph ) )
4		orc 400	. . . 4 |- ( ( x = y /\ ph ) -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
5	1, 3, 4	syl6an 568	. . 3 |- ( A. x x = y -> ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) )
6		sb4 2356	. . . 4 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
7		olc 399	. . . 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 176	. 2 |- ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
10		sbequ1 2110	. . . 4 |- ( x = y -> ( ph -> [ y / x ] ph ) )
11	10	imp 445	. . 3 |- ( ( x = y /\ ph ) -> [ y / x ] ph )
12		sb2 2352	. . 3 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
13	11, 12	jaoi 394	. 2 |- ( ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) -> [ y / x ] ph )
14	9, 13	impbii 199	1 |- ( [ y / x ] ph <-> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   [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-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  dfsb3  2374