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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 8804 | . 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 1536 / cqs 8742 |
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-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ec 8745 df-qs 8749 |
This theorem is referenced by: prjspval 42589 |
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