Theorem qseq12d 43384

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 8785	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   / cqs 8719

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8722  df-qs 8726

This theorem is used by:  prjspval  43724