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Mirrors > Home > ILE Home > Th. List > sbcth | GIF version |
Description: A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.) |
Ref | Expression |
---|---|
sbcth.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
sbcth | ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcth.1 | . . 3 ⊢ 𝜑 | |
2 | 1 | ax-gen 1338 | . 2 ⊢ ∀𝑥𝜑 |
3 | spsbc 2775 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | |
4 | 2, 3 | mpi 15 | 1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 ∈ wcel 1393 [wsbc 2764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 df-sbc 2765 |
This theorem is referenced by: rabrsndc 3438 iota4an 4886 |
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