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Metamath Proof Explorer

Theorem 2sb8e

Description: An equivalent expression for double existence. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring more disjoint variables, but fewer axioms, see 2sb8ef . (Contributed by Wolf Lammen, 2-Nov-2019) (New usage is discouraged.)

Ref Expression
Assertion 2sb8e 
|- ( E. x E. y ph <-> E. z E. w [ z / x ] [ w / y ] ph )

Proof

Step Hyp Ref Expression
1 nfv 
 |-  F/ w ph
2 1 sb8e 
 |-  ( E. y ph <-> E. w [ w / y ] ph )
3 2 exbii 
 |-  ( E. x E. y ph <-> E. x E. w [ w / y ] ph )
4 excom 
 |-  ( E. x E. w [ w / y ] ph <-> E. w E. x [ w / y ] ph )
5 3 4 bitri 
 |-  ( E. x E. y ph <-> E. w E. x [ w / y ] ph )
6 nfv 
 |-  F/ z ph
7 6 nfsb 
 |-  F/ z [ w / y ] ph
8 7 sb8e 
 |-  ( E. x [ w / y ] ph <-> E. z [ z / x ] [ w / y ] ph )
9 8 exbii 
 |-  ( E. w E. x [ w / y ] ph <-> E. w E. z [ z / x ] [ w / y ] ph )
10 excom 
 |-  ( E. w E. z [ z / x ] [ w / y ] ph <-> E. z E. w [ z / x ] [ w / y ] ph )
11 5 9 10 3bitri 
 |-  ( E. x E. y ph <-> E. z E. w [ z / x ] [ w / y ] ph )