Theorem qseq12d 42731

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 8705	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
4	1, 2, 3	syl2anc 590	1 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Syntax hints:   → wi 4   = wceq 1547   / cqs 8639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8642  df-qs 8646

This theorem is referenced by:  prjspval  43060