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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 5648 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | |
2 | soeq2 5611 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) | |
3 | 1, 2 | anbi12d 632 | . 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵))) |
4 | df-we 5634 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
5 | df-we 5634 | . 2 ⊢ (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Or wor 5588 Fr wfr 5629 We wwe 5631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-in 3956 df-ss 3966 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 |
This theorem is referenced by: ordeq 6372 0we1 8506 oieq2 9508 hartogslem1 9537 wemapwe 9692 ween 10030 dfac8 10130 weth 10490 fpwwe2cbv 10625 fpwwe2lem2 10627 fpwwe2lem4 10629 fpwwecbv 10639 fpwwelem 10640 canthwelem 10645 canthwe 10646 pwfseqlem4a 10656 pwfseqlem4 10657 ltweuz 13926 ltwenn 13927 bpolylem 15992 ltbwe 21599 vitali 25130 weeq12d 41782 aomclem6 41801 omeiunle 45233 |
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