|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sbceq1dd | Structured version Visualization version GIF version | ||
| Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) | 
| Ref | Expression | 
|---|---|
| sbceq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| sbceq1dd.2 | ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | 
| Ref | Expression | 
|---|---|
| sbceq1dd | ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbceq1dd.2 | . 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | |
| 2 | sbceq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | 2 | sbceq1d 3792 | . 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) | 
| 4 | 1, 3 | mpbid 232 | 1 ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 [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-ex 1779 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: prmind2 16723 sdclem2 37750 sbceq1ddi 38131 | 
| Copyright terms: Public domain | W3C validator |