Theorem dmqseqeq1i 35911

Description: Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)

Hypothesis

Ref	Expression
dmqseqeq1i.1	⊢ 𝑅 = 𝑆

Assertion

Ref	Expression
dmqseqeq1i	⊢ ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)

Proof of Theorem dmqseqeq1i

Step	Hyp	Ref	Expression
1		dmqseqeq1i.1	. 2 ⊢ 𝑅 = 𝑆
2		dmqseqeq1 35910	. 2 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
3	1, 2	ax-mp 5	1 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1537  dom cdm 5541   / cqs 8274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-br 5053  df-opab 5115  df-cnv 5549  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-ec 8277  df-qs 8281

This theorem is referenced by:  dmqs1cosscnvepreseq  35928