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| 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 5653 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | |
| 2 | soeq2 5614 | . . 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 5639 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
| 5 | df-we 5639 | . 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 206 ∧ wa 395 = wceq 1540 Or wor 5591 Fr wfr 5634 We wwe 5636 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ral 3062 df-ss 3968 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 | 
| This theorem is referenced by: weeq12d 5674 ordeq 6391 0we1 8544 oieq2 9553 wemapwe 9737 ween 10075 dfac8 10176 weth 10535 pwfseqlem4a 10701 pwfseqlem4 10702 ltweuz 14002 ltwenn 14003 bpolylem 16084 ltbwe 22062 vitali 25648 aomclem6 43071 omeiunle 46532 | 
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