Theorem dmqseqeq1 36863

Description: Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)

Assertion

Ref	Expression
dmqseqeq1	⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))

Proof of Theorem dmqseqeq1

Step	Hyp	Ref	Expression
1		dmqseq 36860	. 2 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
2	1	eqeq1d 2738	1 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  dom cdm 5600   / cqs 8528

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3341  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-cnv 5608  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-ec 8531  df-qs 8535

This theorem is referenced by:  dmqseqeq1i  36864  dmqseqeq1d  36865  erALTVeq1  36889  disjdmqseqeq1  36957