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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 4143 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | |
3 | 1, 2 | mpan 424 | . 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
4 | csbexa.2 | . 2 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | mpg 1461 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∀wal 1361 ∈ wcel 2158 Vcvv 2749 ⦋csb 3069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-sbc 2975 df-csb 3070 |
This theorem is referenced by: dfmpo 6237 |
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