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Mirrors > Home > MPE Home > Th. List > freq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.) |
Ref | Expression |
---|---|
freq2 | ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 3621 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | frss 5005 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Fr 𝐴 → 𝑅 Fr 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 → 𝑅 Fr 𝐵)) |
4 | eqimss 3620 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | frss 5005 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴)) |
7 | 3, 6 | impbid 201 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ⊆ wss 3540 Fr wfr 4994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-fr 4997 |
This theorem is referenced by: weeq2 5027 frsn 5112 f1oweALT 7043 frfi 8090 freq12d 36627 ifr0 37675 |
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