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Metamath Proof Explorer

Theorem cbveu

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbveuw , cbveuvw when possible. (Contributed by NM, 25-Nov-1994) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses cbveu.1 
|- F/ y ph
cbveu.2 
|- F/ x ps
cbveu.3 
|- ( x = y -> ( ph <-> ps ) )
Assertion cbveu 
|- ( E! x ph <-> E! y ps )

Proof

Step Hyp Ref Expression
1 cbveu.1 
 |-  F/ y ph
2 cbveu.2 
 |-  F/ x ps
3 cbveu.3 
 |-  ( x = y -> ( ph <-> ps ) )
4 1 sb8eu 
 |-  ( E! x ph <-> E! y [ y / x ] ph )
5 2 3 sbie 
 |-  ( [ y / x ] ph <-> ps )
6 5 eubii 
 |-  ( E! y [ y / x ] ph <-> E! y ps )
7 4 6 bitri 
 |-  ( E! x ph <-> E! y ps )