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Metamath Proof Explorer

Theorem optocl

Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995) Shorten and reduce axiom usage. (Revised by TM, 29-Dec-2025)

Ref Expression
Hypotheses optocl.1 
|- D = ( B X. C )
optocl.2 
|- ( <. x , y >. = A -> ( ph <-> ps ) )
optocl.3 
|- ( ( x e. B /\ y e. C ) -> ph )
Assertion optocl 
|- ( A e. D -> ps )

Proof

Step Hyp Ref Expression
1 optocl.1 
 |-  D = ( B X. C )
2 optocl.2 
 |-  ( <. x , y >. = A -> ( ph <-> ps ) )
3 optocl.3 
 |-  ( ( x e. B /\ y e. C ) -> ph )
4 elxpi 
 |-  ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
5 2 eqcoms 
 |-  ( A = <. x , y >. -> ( ph <-> ps ) )
6 3 5 imbitrid 
 |-  ( A = <. x , y >. -> ( ( x e. B /\ y e. C ) -> ps ) )
7 6 imp 
 |-  ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) -> ps )
8 7 exlimivv 
 |-  ( E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) -> ps )
9 4 8 syl 
 |-  ( A e. ( B X. C ) -> ps )
10 9 1 eleq2s 
 |-  ( A e. D -> ps )