Theorem dmqseqeq1i 38681

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 38680	. 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 206   = wceq 1541  dom cdm 5611   / cqs 8616

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ec 8619  df-qs 8623

This theorem is referenced by:  dmqs1cosscnvepreseq  38700