Theorem dmqseqeq1 39226

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 39223	. 2 ⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
2	1	eqeq1d 2764	1 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560  dom cdm 5647   / cqs 8677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-qs 8684

This theorem is referenced by:  dmqseqeq1i  39227  dmqseqeq1d  39228  erALTVeq1  39253  disjdmqseqeq1  39336