|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > eqsbc1 | Structured version Visualization version GIF version | ||
| Description: Substitution for the left-hand side in an equality. Class version of eqsb1 2866. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2376. (Revised by Wolf Lammen, 29-Apr-2023.) | 
| Ref | Expression | 
|---|---|
| eqsbc1 | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfsbcq 3789 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝐴 / 𝑥]𝑥 = 𝐵)) | |
| 2 | eqeq1 2740 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
| 3 | sbsbc 3791 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝑦 / 𝑥]𝑥 = 𝐵) | |
| 4 | eqsb1 2866 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) | |
| 5 | 3, 4 | bitr3i 277 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) | 
| 6 | 1, 2, 5 | vtoclbg 3556 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 [wsb 2063 ∈ wcel 2107 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: sbceqalOLD 3851 eqsbc2 3853 fmptsnd 7190 fvmptnn04if 22856 snfil 23873 f1omptsnlem 37338 mptsnunlem 37340 topdifinffinlem 37349 relowlpssretop 37366 iotavalb 44454 onfrALTlem5 44567 eqsbc2VD 44865 onfrALTlem5VD 44910 | 
| Copyright terms: Public domain | W3C validator |