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Metamath Proof Explorer

Theorem cbv2h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 11-May-1993) (New usage is discouraged.)

Ref Expression
Hypotheses cbv2h.1 
|- ( ph -> ( ps -> A. y ps ) )
cbv2h.2 
|- ( ph -> ( ch -> A. x ch ) )
cbv2h.3 
|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
Assertion cbv2h 
|- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

Proof

Step Hyp Ref Expression
1 cbv2h.1 
 |-  ( ph -> ( ps -> A. y ps ) )
2 cbv2h.2 
 |-  ( ph -> ( ch -> A. x ch ) )
3 cbv2h.3 
 |-  ( ph -> ( x = y -> ( ps <-> ch ) ) )
4 biimp 
 |-  ( ( ps <-> ch ) -> ( ps -> ch ) )
5 3 4 syl6 
 |-  ( ph -> ( x = y -> ( ps -> ch ) ) )
6 1 2 5 cbv1h 
 |-  ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
7 equcomi 
 |-  ( y = x -> x = y )
8 biimpr 
 |-  ( ( ps <-> ch ) -> ( ch -> ps ) )
9 7 3 8 syl56 
 |-  ( ph -> ( y = x -> ( ch -> ps ) ) )
10 2 1 9 cbv1h 
 |-  ( A. y A. x ph -> ( A. y ch -> A. x ps ) )
11 10 alcoms 
 |-  ( A. x A. y ph -> ( A. y ch -> A. x ps ) )
12 6 11 impbid 
 |-  ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )