Theorem freq12d

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

	Ref	Expression
Hypotheses	freq12d.1	|- ( ph -> R = S )
	freq12d.2	|- ( ph -> A = B )
Assertion	freq12d	|- ( ph -> ( R Fr A <-> S Fr B ) )

Proof

Step	Hyp	Ref	Expression
1		freq12d.1	|- ( ph -> R = S )
2		freq12d.2	|- ( ph -> A = B )
3		freq1	|- ( R = S -> ( R Fr A <-> S Fr A ) )
4		freq2	|- ( A = B -> ( S Fr A <-> S Fr B ) )
5	3 4	sylan9bb	|- ( ( R = S /\ A = B ) -> ( R Fr A <-> S Fr B ) )
6	1 2 5	syl2anc	|- ( ph -> ( R Fr A <-> S Fr B ) )