|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > preqr2 | Structured version Visualization version GIF version | ||
| Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.) | 
| Ref | Expression | 
|---|---|
| preqr1.a | ⊢ 𝐴 ∈ V | 
| preqr1.b | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| preqr2 | ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | prcom 4732 | . . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶} | |
| 2 | prcom 4732 | . . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶} | |
| 3 | 1, 2 | eqeq12i 2755 | . 2 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶}) | 
| 4 | preqr1.a | . . 3 ⊢ 𝐴 ∈ V | |
| 5 | preqr1.b | . . 3 ⊢ 𝐵 ∈ V | |
| 6 | 4, 5 | preqr1 4848 | . 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵) | 
| 7 | 3, 6 | sylbi 217 | 1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {cpr 4628 | 
| 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-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: preq12b 4850 opth 5481 opthreg 9658 usgredgreu 29235 uspgredg2vtxeu 29237 altopthsn 35962 | 
| Copyright terms: Public domain | W3C validator |