Theorem qseq1d 35551

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
qseq1d	⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))

Proof of Theorem qseq1d

Step	Hyp	Ref	Expression
1		qseq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		qseq1 8346	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))

Syntax hints:   → wi 4   = wceq 1536   / cqs 8291

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-rex 3147  df-qs 8298

This theorem is referenced by:  n0el3  35889