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Metamath Proof Explorer

Theorem iotaval

Description: Theorem 8.19 in Quine p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011) Remove dependency on ax-10 , ax-11 , ax-12 . (Revised by SN, 23-Nov-2024)

Ref Expression
Assertion iotaval 
|- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Proof

Step Hyp Ref Expression
1 abbi 
 |-  ( A. x ( ph <-> x = y ) -> { x | ph } = { x | x = y } )
2 df-sn 
 |-  { y } = { x | x = y }
3 1 2 eqtr4di 
 |-  ( A. x ( ph <-> x = y ) -> { x | ph } = { y } )
4 iotaval2 
 |-  ( { x | ph } = { y } -> ( iota x ph ) = y )
5 3 4 syl 
 |-  ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )