Theorem qseq2d 8737

Description: Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)

Hypothesis

Ref	Expression
qseq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
qseq2d	⊢ (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵))

Proof of Theorem qseq2d

Step	Hyp	Ref	Expression
1		qseq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		qseq2 8734	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵))

Syntax hints:   → wi 4   = wceq 1540   / cqs 8673

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ec 8676  df-qs 8680

This theorem is referenced by:  qustriv  33342  opprqusbas  33466  qsdrngi  33473  pstmval  33892  prjspnval2  42613  0prjspn  42623