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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 5670 | . . 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 5671 | . . 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 278 | 1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 We wwe 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-v 3475 df-in 3956 df-ss 3966 df-br 5153 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 |
This theorem is referenced by: fnwe2lem1 42505 aomclem1 42509 aomclem4 42512 aomclem5 42513 aomclem6 42514 |
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