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	$$⊢\phi \to R=S$$
	freq12d.2	$$⊢\phi \to A=B$$
Assertion	freq12d	$$⊢\phi \to \left(R\mathrm{Fr}A\leftrightarrow S\mathrm{Fr}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		freq12d.1	$$⊢\phi \to R=S$$
2		freq12d.2	$$⊢\phi \to A=B$$
3		freq1	$$⊢R=S\to \left(R\mathrm{Fr}A\leftrightarrow S\mathrm{Fr}A\right)$$
4		freq2	$$⊢A=B\to \left(S\mathrm{Fr}A\leftrightarrow S\mathrm{Fr}B\right)$$
5	3 4	sylan9bb	$$⊢\left(R=S\wedge A=B\right)\to \left(R\mathrm{Fr}A\leftrightarrow S\mathrm{Fr}B\right)$$
6	1 2 5	syl2anc	$$⊢\phi \to \left(R\mathrm{Fr}A\leftrightarrow S\mathrm{Fr}B\right)$$