Theorem qseq12 8699

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 8694	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
2		qseq2 8695	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 / 𝐶) = (𝐵 / 𝐷))
3	1, 2	sylan9eq 2791	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   / cqs 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-qs 8641

This theorem is referenced by:  dmqseq  38894  qseq12d  42491