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Metamath Proof Explorer
Description: Equality theorem for domain quotient set, deduction version.
(Contributed by Peter Mazsa, 26-Sep-2021)
|
|
Ref |
Expression |
|
Hypothesis |
dmqseqeq1d.1 |
⊢ ( 𝜑 → 𝑅 = 𝑆 ) |
|
Assertion |
dmqseqeq1d |
⊢ ( 𝜑 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dmqseqeq1d.1 |
⊢ ( 𝜑 → 𝑅 = 𝑆 ) |
| 2 |
|
dmqseqeq1 |
⊢ ( 𝑅 = 𝑆 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → ( ( dom 𝑅 / 𝑅 ) = 𝐴 ↔ ( dom 𝑆 / 𝑆 ) = 𝐴 ) ) |