Theorem weeq12d

Description: Equality deduction for well-orderings. (Contributed by Stefan O'Rear, 19-Jan-2015) (Proof shortened by Matthew House, 10-Sep-2025)

	Ref	Expression
Hypotheses	weeq12d.1	|- ( ph -> R = S )
	weeq12d.2	|- ( ph -> A = B )
Assertion	weeq12d	|- ( ph -> ( R We A <-> S We B ) )

Proof

Step	Hyp	Ref	Expression
1		weeq12d.1	|- ( ph -> R = S )
2		weeq12d.2	|- ( ph -> A = B )
3		weeq1	|- ( R = S -> ( R We A <-> S We A ) )
4		weeq2	|- ( A = B -> ( S We A <-> S We B ) )
5	3 4	sylan9bb	|- ( ( R = S /\ A = B ) -> ( R We A <-> S We B ) )
6	1 2 5	syl2anc	|- ( ph -> ( R We A <-> S We B ) )