Theorem freq12d 5632

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 5630	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
4		freq2 5631	. . 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))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   Fr wfr 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-ss 3923  df-br 5111  df-fr 5616

This theorem is referenced by:  weiunfr  36959