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Metamath Proof Explorer

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 ) ) )