Theorem qseq12d 42212

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 8696	. 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 1540   / cqs 8631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8634  df-qs 8638

This theorem is referenced by:  prjspval  42576