Theorem dfsb2 2055

Description: An alternate definition of proper substitution that, like df-sb 1649, 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 1747	. . . 4 |- (A.x x = y -> x = y)
2		sbequ2 1650	. . . . 5 |- (x = y -> ([y / x]ph -> ph))
3	2	sps 1754	. . . 4 |- (A.x x = y -> ([y / x]ph -> ph))
4		orc 374	. . . 4 |- ((x = y /\ ph) -> ((x = y /\ ph) \/ A.x(x = y -> ph)))
5	1, 3, 4	ee12an 1363	. . 3 |- (A.x x = y -> ([y / x]ph -> ((x = y /\ ph) \/ A.x(x = y -> ph))))
6		sb4 2053	. . . 4 |- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))
7		olc 373	. . . 4 |- (A.x(x = y -> ph) -> ((x = y /\ ph) \/ A.x(x = y -> ph)))
8	6, 7	syl6 29	. . 3 |- (-. A.x x = y -> ([y / x]ph -> ((x = y /\ ph) \/ A.x(x = y -> ph))))
9	5, 8	pm2.61i 156	. 2 |- ([y / x]ph -> ((x = y /\ ph) \/ A.x(x = y -> ph)))
10		sbequ1 1918	. . . 4 |- (x = y -> (ph -> [y / x]ph))
11	10	imp 418	. . 3 |- ((x = y /\ ph) -> [y / x]ph)
12		sb2 2023	. . 3 |- (A.x(x = y -> ph) -> [y / x]ph)
13	11, 12	jaoi 368	. 2 |- (((x = y /\ ph) \/ A.x(x = y -> ph)) -> [y / x]ph)
14	9, 13	impbii 180	1 |- ([y / x]ph <-> ((x = y /\ ph) \/ A.x(x = y -> ph)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  dfsb3  2056