Theorem freq1 4489

Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)

Assertion

Ref	Expression
freq1	⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))

Proof of Theorem freq1

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frforeq1 4488	. . 3 ⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑆𝐴𝑠))
2	1	albidv 1877	. 2 ⊢ (𝑅 = 𝑆 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠))
3		df-frind 4477	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
4		df-frind 4477	. 2 ⊢ (𝑆 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠)
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105  ∀wal 1400   = wceq 1402   FrFor wfrfor 4472   Fr wfr 4473

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-clel 2234  df-ral 2533  df-br 4131  df-frfor 4476  df-frind 4477

This theorem is used by:  weeq1  4501