Theorem freq12d 5629

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

Hypotheses

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

Assertion

Ref	Expression
freq12d	⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Proof of Theorem freq12d

Step	Hyp	Ref	Expression
1		freq12d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		freq12d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		freq1 5627	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
4		freq2 5628	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
5	3, 4	sylan9bb 518	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
6	1, 2, 5	syl2anc 595	1 ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   Fr wfr 5610

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-ss 3921  df-br 5109  df-fr 5613

This theorem is used by:  weiunfr  37006