|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sbcth2 | Structured version Visualization version GIF version | ||
| Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) | 
| Ref | Expression | 
|---|---|
| sbcth2.1 | ⊢ (𝑥 ∈ 𝐵 → 𝜑) | 
| Ref | Expression | 
|---|---|
| sbcth2 | ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbcth2.1 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
| 2 | 1 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑 | 
| 3 | rspsbc 3879 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑)) | |
| 4 | 2, 3 | mpi 20 | 1 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 [wsbc 3788 | 
| 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-11 2157 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-sbc 3789 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |