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Metamath Proof Explorer

Theorem ifeq12

Description: Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004)

Ref Expression
Assertion ifeq12 
|- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )

Proof

Step Hyp Ref Expression
1 ifeq1 
 |-  ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
2 ifeq2 
 |-  ( C = D -> if ( ph , B , C ) = if ( ph , B , D ) )
3 1 2 sylan9eq 
 |-  ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )