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| 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 3062 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑 | 
| 3 | rspsbc 3878 | . 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 2107 ∀wral 3060 [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-11 2156 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-ral 3061 df-sbc 3788 | 
| This theorem is referenced by: (None) | 
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