Theorem bj-cleq

Description: Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019)

	Ref	Expression
Assertion	bj-cleq	|- ( A = B -> { x | { x } e. ( A " C ) } = { x | { x } e. ( B " C ) } )

Proof

Step	Hyp	Ref	Expression
1		imaeq1	|- ( A = B -> ( A " C ) = ( B " C ) )
2		eleq2	|- ( ( A " C ) = ( B " C ) -> ( { x } e. ( A " C ) <-> { x } e. ( B " C ) ) )
3	2	alrimiv	|- ( ( A " C ) = ( B " C ) -> A. x ( { x } e. ( A " C ) <-> { x } e. ( B " C ) ) )
4		abbi1	|- ( A. x ( { x } e. ( A " C ) <-> { x } e. ( B " C ) ) -> { x | { x } e. ( A " C ) } = { x | { x } e. ( B " C ) } )
5	1 3 4	3syl	|- ( A = B -> { x | { x } e. ( A " C ) } = { x | { x } e. ( B " C ) } )