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	⊢ ( 𝜑 → 𝑅 = 𝑆 )
	freq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	freq12d	⊢ ( 𝜑 → ( 𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		freq12d.1	⊢ ( 𝜑 → 𝑅 = 𝑆 )
2		freq12d.2	⊢ ( 𝜑 → 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	⊢ ( 𝜑 → ( 𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵 ) )