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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 4115 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | |
3 | 1, 2 | mpan 422 | . 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
4 | csbexa.2 | . 2 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | mpg 1444 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∀wal 1346 ∈ wcel 2141 Vcvv 2730 ⦋csb 3049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-sbc 2956 df-csb 3050 |
This theorem is referenced by: dfmpo 6199 |
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