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Mirrors > Home > MPE Home > Th. List > weeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.) |
Ref | Expression |
---|---|
weeq2 | ⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | freq2 5009 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | |
2 | soeq2 4979 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) | |
3 | 1, 2 | anbi12d 743 | . 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵))) |
4 | df-we 4999 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
5 | df-we 4999 | . 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 302 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Or wor 4958 Fr wfr 4994 We wwe 4996 |
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-ral 2901 df-in 3547 df-ss 3554 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 |
This theorem is referenced by: ordeq 5647 0we1 7473 oieq2 8301 hartogslem1 8330 wemapwe 8477 ween 8741 dfac8 8840 weth 9200 fpwwe2cbv 9331 fpwwe2lem2 9333 fpwwe2lem5 9335 fpwwecbv 9345 fpwwelem 9346 canthwelem 9351 canthwe 9352 pwfseqlem4a 9362 pwfseqlem4 9363 ltweuz 12622 ltwenn 12623 bpolylem 14618 ltbwe 19293 vitali 23188 weeq12d 36628 aomclem6 36647 omeiunle 39407 |
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