Theorem dmqseqd 39416

Description: Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)

Hypothesis

Ref	Expression
dmqseqd.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
dmqseqd	⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

Proof of Theorem dmqseqd

Step	Hyp	Ref	Expression
1		dmqseqd.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		dmqseq 39414	. 2 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
3	1, 2	syl 18	1 ⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  dom cdm 5666   / cqs 8702

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8705  df-qs 8709

This theorem is used by:  releldmqs  39433  releldmqscoss  39435