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Mirrors > Home > ILE Home > Th. List > freq1 | GIF version |
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.) |
Ref | Expression |
---|---|
freq1 | ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frforeq1 4315 | . . 3 ⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑆𝐴𝑠)) | |
2 | 1 | albidv 1811 | . 2 ⊢ (𝑅 = 𝑆 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠)) |
3 | df-frind 4304 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠) | |
4 | df-frind 4304 | . 2 ⊢ (𝑆 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑆𝐴𝑠) | |
5 | 2, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 = wceq 1342 FrFor wfrfor 4299 Fr wfr 4300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-cleq 2157 df-clel 2160 df-ral 2447 df-br 3977 df-frfor 4303 df-frind 4304 |
This theorem is referenced by: weeq1 4328 |
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