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 )