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Mirrors > Home > MPE Home > Th. List > qseq12 | Structured version Visualization version GIF version |
Description: Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.) |
Ref | Expression |
---|---|
qseq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qseq1 8346 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | |
2 | qseq2 8347 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 / 𝐶) = (𝐵 / 𝐷)) | |
3 | 1, 2 | sylan9eq 2879 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 / cqs 8291 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ec 8294 df-qs 8298 |
This theorem is referenced by: dmqseq 35879 qseq12d 39130 |
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