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Theorem qseq12d 39440
 Description: Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
Hypotheses
Ref Expression
qseq12d.1 (𝜑𝐴 = 𝐵)
qseq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
qseq12d (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Proof of Theorem qseq12d
StepHypRef Expression
1 qseq12d.1 . 2 (𝜑𝐴 = 𝐵)
2 qseq12d.2 . 2 (𝜑𝐶 = 𝐷)
3 qseq12 8333 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
41, 2, 3syl2anc 587 1 (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   / cqs 8274 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ec 8277  df-qs 8281 This theorem is referenced by:  prjspval  39640
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