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Theorem dmqseqeq1i 37135
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
StepHypRef Expression
1 dmqseqeq1i.1 . 2 𝑅 = 𝑆
2 dmqseqeq1 37134 . 2 (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
31, 2ax-mp 5 1 ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  dom cdm 5638   / cqs 8654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8657  df-qs 8661
This theorem is referenced by:  dmqs1cosscnvepreseq  37153
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