|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sbcbi1 | Structured version Visualization version GIF version | ||
| Description: Distribution of class substitution over biconditional. One direction of sbcbig 3839 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) | 
| Ref | Expression | 
|---|---|
| sbcbi1 | ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbcex 3797 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → 𝐴 ∈ V) | |
| 2 | sbcbig 3839 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | |
| 3 | 2 | biimpd 229 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | 
| 4 | 1, 3 | mpcom 38 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 Vcvv 3479 [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-10 2140 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-sbc 3788 | 
| This theorem is referenced by: dfconngr1 30208 | 
| Copyright terms: Public domain | W3C validator |