Theorem seeq12d

Description: Equality deduction for the set-like predicate. (Contributed by Matthew House, 10-Sep-2025)

	Ref	Expression
Hypotheses	seeq12d.1	$$⊢\phi \to R=S$$
	seeq12d.2	$$⊢\phi \to A=B$$
Assertion	seeq12d	$$⊢\phi \to \left(R\mathrm{Se}A\leftrightarrow S\mathrm{Se}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		seeq12d.1	$$⊢\phi \to R=S$$
2		seeq12d.2	$$⊢\phi \to A=B$$
3		seeq1	$$⊢R=S\to \left(R\mathrm{Se}A\leftrightarrow S\mathrm{Se}A\right)$$
4		seeq2	$$⊢A=B\to \left(S\mathrm{Se}A\leftrightarrow S\mathrm{Se}B\right)$$
5	3 4	sylan9bb	$$⊢\left(R=S\wedge A=B\right)\to \left(R\mathrm{Se}A\leftrightarrow S\mathrm{Se}B\right)$$
6	1 2 5	syl2anc	$$⊢\phi \to \left(R\mathrm{Se}A\leftrightarrow S\mathrm{Se}B\right)$$