Theorem qseq12 8727

Description: Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.)

Assertion

Ref	Expression
qseq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Proof of Theorem qseq12

Step	Hyp	Ref	Expression
1		qseq1 8722	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
2		qseq2 8723	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 / 𝐶) = (𝐵 / 𝐷))
3	1, 2	sylan9eq 2807	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   / cqs 8661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-cnv 5644  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-ec 8664  df-qs 8668

This theorem is referenced by:  dmqseq  39161  qseq12d  42794