Theorem dmqseqeq1d 38636

Description: Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.)

Hypothesis

Ref	Expression
dmqseqeq1d.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
dmqseqeq1d	⊢ (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))

Proof of Theorem dmqseqeq1d

Step	Hyp	Ref	Expression
1		dmqseqeq1d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		dmqseqeq1 38634	. 2 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  dom cdm 5638   / cqs 8670

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673  df-qs 8677

This theorem is referenced by: (None)