|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > qseq1d | Structured version Visualization version GIF version | ||
| Description: Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.) | 
| Ref | Expression | 
|---|---|
| qseq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| qseq1d | ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | qseq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | qseq1 8801 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 / cqs 8744 | 
| 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 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-rex 3071 df-qs 8751 | 
| This theorem is referenced by: fracbas 33307 n0elim 38651 | 
| Copyright terms: Public domain | W3C validator |