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| Mirrors > Home > MPE Home > Th. List > weeq12d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for well-orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.) (Proof shortened by Matthew House, 10-Sep-2025.) | 
| Ref | Expression | 
|---|---|
| weeq12d.1 | ⊢ (𝜑 → 𝑅 = 𝑆) | 
| weeq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| weeq12d | ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | weeq12d.1 | . 2 ⊢ (𝜑 → 𝑅 = 𝑆) | |
| 2 | weeq12d.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | weeq1 5671 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴)) | |
| 4 | weeq2 5672 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑆 We 𝐴 ↔ 𝑆 We 𝐵)) | |
| 5 | 3, 4 | sylan9bb 509 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) | 
| 6 | 1, 2, 5 | syl2anc 584 | 1 ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 We wwe 5635 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-ex 1779 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-ss 3967 df-br 5143 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 | 
| This theorem is referenced by: hartogslem1 9583 fpwwe2cbv 10671 fpwwe2lem2 10673 fpwwe2lem4 10675 fpwwecbv 10685 fpwwelem 10686 canthwelem 10691 canthwe 10692 pwfseqlem4 10703 fnwe2lem1 43067 aomclem1 43071 aomclem4 43074 aomclem5 43075 aomclem6 43076 | 
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