Theorem dfsb2

Description: An alternate definition of proper substitution that, like dfsb1 , mixes free and bound variables to avoid distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 17-Feb-2005) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sp	|- ( A. x x = y -> x = y )
2		sbequ2	|- ( x = y -> ( [ y / x ] ph -> ph ) )
3	2	sps	|- ( A. x x = y -> ( [ y / x ] ph -> ph ) )
4		orc	|- ( ( x = y /\ ph ) -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
5	1 3 4	syl6an	|- ( A. x x = y -> ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) )
6		sb4b	|- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )
7		olc	|- ( A. x ( x = y -> ph ) -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
8	6 7	biimtrdi	|- ( -. A. x x = y -> ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) )
9	5 8	pm2.61i	|- ( [ y / x ] ph -> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
10		sbequ1	|- ( x = y -> ( ph -> [ y / x ] ph ) )
11	10	imp	|- ( ( x = y /\ ph ) -> [ y / x ] ph )
12		sb2	|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
13	11 12	jaoi	|- ( ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) -> [ y / x ] ph )
14	9 13	impbii	|- ( [ y / x ] ph <-> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )