Theorem freq12d 5587

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 5585	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
4		freq2 5586	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
5	3, 4	sylan9bb 514	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
6	1, 2, 5	syl2anc 590	1 ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   Fr wfr 5568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-ss 3900  df-br 5073  df-fr 5571

This theorem is referenced by:  weiunfr  36695