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Metamath Proof Explorer

Theorem cbvmo

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 cbvmow , cbvmovw when possible. (Contributed by NM, 9-Mar-1995) (Revised by Andrew Salmon, 8-Jun-2011) (Proof shortened by Wolf Lammen, 4-Jan-2023) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 cbvmo.1 
 |-  F/ y ph
2 cbvmo.2 
 |-  F/ x ps
3 cbvmo.3 
 |-  ( x = y -> ( ph <-> ps ) )
4 1 sb8mo 
 |-  ( E* x ph <-> E* y [ y / x ] ph )
5 2 3 sbie 
 |-  ( [ y / x ] ph <-> ps )
6 5 mobii 
 |-  ( E* y [ y / x ] ph <-> E* y ps )
7 4 6 bitri 
 |-  ( E* x ph <-> E* y ps )