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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > weeq12d | Structured version Visualization version GIF version |
Description: Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Ref | Expression |
---|---|
weeq12d.l | ⊢ (𝜑 → 𝑅 = 𝑆) |
weeq12d.r | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
weeq12d | ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | weeq12d.l | . . 3 ⊢ (𝜑 → 𝑅 = 𝑆) | |
2 | weeq1 5343 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴)) |
4 | weeq12d.r | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | weeq2 5344 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵)) |
7 | 3, 6 | bitrd 271 | 1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 We wwe 5313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-ral 3094 df-rex 3095 df-in 3798 df-ss 3805 df-br 4887 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 |
This theorem is referenced by: fnwe2lem1 38572 aomclem1 38576 aomclem4 38579 aomclem5 38580 aomclem6 38581 |
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