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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 5490 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | |
2 | soeq2 5459 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) | |
3 | 1, 2 | anbi12d 634 | . 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵))) |
4 | df-we 5480 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
5 | df-we 5480 | . 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 317 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 Or wor 5437 Fr wfr 5475 We wwe 5477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-v 3399 df-in 3848 df-ss 3858 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 |
This theorem is referenced by: ordeq 6173 0we1 8155 oieq2 9043 hartogslem1 9072 wemapwe 9226 ween 9528 dfac8 9628 weth 9988 fpwwe2cbv 10123 fpwwe2lem2 10125 fpwwe2lem4 10127 fpwwecbv 10137 fpwwelem 10138 canthwelem 10143 canthwe 10144 pwfseqlem4a 10154 pwfseqlem4 10155 ltweuz 13413 ltwenn 13414 bpolylem 15487 ltbwe 20848 vitali 24358 weeq12d 40421 aomclem6 40440 omeiunle 43581 |
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