Theorem dfsb2 1223

Description: An alternate definition of proper substitution that, like df-sb 1170, mixes free and bound variables to avoid distinct variable requirements.

Assertion

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

Proof of Theorem dfsb2

Step	Hyp	Ref	Expression
1		sbequ2 1177	. . . . . 6 |- (x = y -> ([y / x]ph -> ph))
2	1	a4s 982	. . . . 5 |- (A.x x = y -> ([y / x]ph -> ph))
3		ax-4 971	. . . . 5 |- (A.x x = y -> x = y)
4	2, 3	jctild 600	. . . 4 |- (A.x x = y -> ([y / x]ph -> (x = y /\ ph)))
5		orc 269	. . . 4 |- ((x = y /\ ph) -> ((x = y /\ ph) \/ A.x(x = y -> ph)))
6	4, 5	syl6 22	. . 3 |- (A.x x = y -> ([y / x]ph -> ((x = y /\ ph) \/ A.x(x = y -> ph))))
7		sb4 1221	. . . 4 |- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))
8		olc 268	. . . 4 |- (A.x(x = y -> ph) -> ((x = y /\ ph) \/ A.x(x = y -> ph)))
9	7, 8	syl6 22	. . 3 |- (-. A.x x = y -> ([y / x]ph -> ((x = y /\ ph) \/ A.x(x = y -> ph))))
10	6, 9	pm2.61i 126	. 2 |- ([y / x]ph -> ((x = y /\ ph) \/ A.x(x = y -> ph)))
11		sbequ1 1176	. . . 4 |- (x = y -> (ph -> [y / x]ph))
12	11	imp 350	. . 3 |- ((x = y /\ ph) -> [y / x]ph)
13		sb2 1175	. . 3 |- (A.x(x = y -> ph) -> [y / x]ph)
14	12, 13	jaoi 341	. 2 |- (((x = y /\ ph) \/ A.x(x = y -> ph)) -> [y / x]ph)
15	10, 14	impbi 157	1 |- ([y / x]ph <-> ((x = y /\ ph) \/ A.x(x = y -> ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 952   = wceq 954  [wsbc 1168

This theorem is referenced by:  dfsb3 1224

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170

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