Theorem dmqseqeq1i 38667

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 38666	. 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 1540  dom cdm 5659   / cqs 8723

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 2708

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 2715  df-cleq 2728  df-clel 2810  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8726  df-qs 8730

This theorem is referenced by:  dmqs1cosscnvepreseq  38685