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Metamath Proof Explorer

Theorem 2sb5nd

Description: Equivalence for double substitution 2sb5 without distinct x , y requirement. 2sb5nd is derived from 2sb5ndVD . (Contributed by Alan Sare, 30-Apr-2014) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 2sb5nd 
|- ( ( -. A. x x = y \/ u = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )

Proof

Step Hyp Ref Expression
1 ax6e2ndeq 
 |-  ( ( -. A. x x = y \/ u = v ) <-> E. x E. y ( x = u /\ y = v ) )
2 anabs5 
 |-  ( ( E. x E. y ( x = u /\ y = v ) /\ ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ) <-> ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
3 2pm13.193 
 |-  ( ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( ( x = u /\ y = v ) /\ ph ) )
4 3 exbii 
 |-  ( E. y ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> E. y ( ( x = u /\ y = v ) /\ ph ) )
5 nfs1v 
 |-  F/ y [ v / y ] ph
6 5 nfsb 
 |-  F/ y [ u / x ] [ v / y ] ph
7 6 19.41 
 |-  ( E. y ( ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
8 4 7 bitr3i 
 |-  ( E. y ( ( x = u /\ y = v ) /\ ph ) <-> ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
9 8 exbii 
 |-  ( E. x E. y ( ( x = u /\ y = v ) /\ ph ) <-> E. x ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
10 nfs1v 
 |-  F/ x [ u / x ] [ v / y ] ph
11 10 19.41 
 |-  ( E. x ( E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) )
12 9 11 bitr2i 
 |-  ( ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) )
13 12 anbi2i 
 |-  ( ( E. x E. y ( x = u /\ y = v ) /\ ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) ) <-> ( E. x E. y ( x = u /\ y = v ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )
14 2 13 bitr3i 
 |-  ( ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. x E. y ( x = u /\ y = v ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )
15 pm5.32 
 |-  ( ( E. x E. y ( x = u /\ y = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) ) <-> ( ( E. x E. y ( x = u /\ y = v ) /\ [ u / x ] [ v / y ] ph ) <-> ( E. x E. y ( x = u /\ y = v ) /\ E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) ) )
16 14 15 mpbir 
 |-  ( E. x E. y ( x = u /\ y = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )
17 1 16 sylbi 
 |-  ( ( -. A. x x = y \/ u = v ) -> ( [ u / x ] [ v / y ] ph <-> E. x E. y ( ( x = u /\ y = v ) /\ ph ) ) )