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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 1393 | . 2 ⊢ ∀𝑥𝜑 |
3 | spsbc 2873 | . 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 1297 ∈ wcel 1448 [wsbc 2862 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-v 2643 df-sbc 2863 |
This theorem is referenced by: rabrsndc 3538 iota4an 5043 |
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