Theorem qseq12d 40214

Description: Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)

Hypotheses

Ref	Expression
qseq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
qseq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
qseq12d	⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Proof of Theorem qseq12d

Step	Hyp	Ref	Expression
1		qseq12d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		qseq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		qseq12 8556	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Syntax hints:   → wi 4   = wceq 1539   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504

This theorem is referenced by:  prjspval  40442