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| Mirrors > Home > MPE Home > Th. List > seeq12d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for the set-like predicate. (Contributed by Matthew House, 10-Sep-2025.) |
| Ref | Expression |
|---|---|
| seeq12d.1 | ⊢ (𝜑 → 𝑅 = 𝑆) |
| seeq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| seeq12d | ⊢ (𝜑 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | seeq12d.1 | . 2 ⊢ (𝜑 → 𝑅 = 𝑆) | |
| 2 | seeq12d.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | seeq1 5655 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) | |
| 4 | seeq2 5656 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Se 𝐴 ↔ 𝑆 Se 𝐵)) | |
| 5 | 3, 4 | sylan9bb 509 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) |
| 6 | 1, 2, 5 | syl2anc 584 | 1 ⊢ (𝜑 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 Se wse 5635 |
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-br 5144 df-se 5638 |
| This theorem is referenced by: (None) |
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