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| Mirrors > Home > MPE Home > Th. List > Mathboxes > qseq12d | Structured version Visualization version GIF version | ||
| Description: Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.) |
| Ref | Expression |
|---|---|
| qseq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| qseq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| qseq12d | ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | qseq12d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | qseq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 3 | qseq12 8806 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 / cqs 8744 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 |
| This theorem is referenced by: prjspval 42613 |
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