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Metamath Proof Explorer

Theorem hblem

Description: Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993) (Revised by Andrew Salmon, 11-Jul-2011) Add disjoint variable condition to avoid ax-13 . See hblemg for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

Ref Expression
Hypothesis hblem.1 
|- ( y e. A -> A. x y e. A )
Assertion hblem 
|- ( z e. A -> A. x z e. A )

Proof

Step Hyp Ref Expression
1 hblem.1 
 |-  ( y e. A -> A. x y e. A )
2 1 hbsbw 
 |-  ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3 clelsb1 
 |-  ( [ z / y ] y e. A <-> z e. A )
4 3 albii 
 |-  ( A. x [ z / y ] y e. A <-> A. x z e. A )
5 2 3 4 3imtr3i 
 |-  ( z e. A -> A. x z e. A )