Theorem unabw

Description: Union of two class abstractions. Version of unab using implicit substitution, which does not require ax-8 , ax-10 , ax-12 . (Contributed by Gino Giotto, 15-Oct-2024)

	Ref	Expression
Hypotheses	unabw.1	|- ( x = y -> ( ph <-> ch ) )
	unabw.2	|- ( x = y -> ( ps <-> th ) )
Assertion	unabw	|- ( { x | ph } u. { x | ps } ) = { y | ( ch \/ th ) }

Proof

Step	Hyp	Ref	Expression
1		unabw.1	|- ( x = y -> ( ph <-> ch ) )
2		unabw.2	|- ( x = y -> ( ps <-> th ) )
3		df-un	|- ( { x | ph } u. { x | ps } ) = { y | ( y e. { x | ph } \/ y e. { x | ps } ) }
4		df-clab	|- ( y e. { x | ph } <-> [ y / x ] ph )
5	1	sbievw	|- ( [ y / x ] ph <-> ch )
6	4 5	bitri	|- ( y e. { x | ph } <-> ch )
7		df-clab	|- ( y e. { x | ps } <-> [ y / x ] ps )
8	2	sbievw	|- ( [ y / x ] ps <-> th )
9	7 8	bitri	|- ( y e. { x | ps } <-> th )
10	6 9	orbi12i	|- ( ( y e. { x | ph } \/ y e. { x | ps } ) <-> ( ch \/ th ) )
11	10	abbii	|- { y | ( y e. { x | ph } \/ y e. { x | ps } ) } = { y | ( ch \/ th ) }
12	3 11	eqtri	|- ( { x | ph } u. { x | ps } ) = { y | ( ch \/ th ) }