Theorem weeq12d

Description: Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015)

	Ref	Expression
Hypotheses	weeq12d.l	$$⊢\phi \to R=S$$
	weeq12d.r	$$⊢\phi \to A=B$$
Assertion	weeq12d	$$⊢\phi \to \left(R\mathrm{We}A\leftrightarrow S\mathrm{We}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		weeq12d.l	$$⊢\phi \to R=S$$
2		weeq12d.r	$$⊢\phi \to A=B$$
3		weeq1	$$⊢R=S\to \left(R\mathrm{We}A\leftrightarrow S\mathrm{We}A\right)$$
4	1 3	syl	$$⊢\phi \to \left(R\mathrm{We}A\leftrightarrow S\mathrm{We}A\right)$$
5		weeq2	$$⊢A=B\to \left(S\mathrm{We}A\leftrightarrow S\mathrm{We}B\right)$$
6	2 5	syl	$$⊢\phi \to \left(S\mathrm{We}A\leftrightarrow S\mathrm{We}B\right)$$
7	4 6	bitrd	$$⊢\phi \to \left(R\mathrm{We}A\leftrightarrow S\mathrm{We}B\right)$$