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Mirrors > Home > ILE Home > Th. List > csbexa | GIF version |
Description: The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
csbexa.1 | ⊢ 𝐴 ∈ V |
csbexa.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
csbexa | ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbexa.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | csbexga 3906 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | |
3 | 1, 2 | mpan 414 | . 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
4 | csbexa.2 | . 2 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | mpg 1380 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∀wal 1282 ∈ wcel 1433 Vcvv 2601 ⦋csb 2908 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-csb 2909 |
This theorem is referenced by: dfmpt2 5864 |
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