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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 5658 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) | |
4 | seeq2 5659 | . . 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 1536 Se wse 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rab 3433 df-v 3479 df-in 3969 df-ss 3979 df-br 5148 df-se 5641 |
This theorem is referenced by: (None) |
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