Theorem freq12d 5654

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 5652	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
4		freq2 5653	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
5	3, 4	sylan9bb 509	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
6	1, 2, 5	syl2anc 584	1 ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   Fr wfr 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-ss 3968  df-br 5144  df-fr 5637

This theorem is referenced by:  weiunfr  36468