Theorem qseq12 8747

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 8742	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
2		qseq2 8743	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 / 𝐶) = (𝐵 / 𝐷))
3	1, 2	sylan9eq 2820	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   / cqs 8681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ec 8684  df-qs 8688

This theorem is referenced by:  dmqseq  39235  qseq12d  42868