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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 4104 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | |
3 | 1, 2 | mpan 421 | . 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) |
4 | csbexa.2 | . 2 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | mpg 1438 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∀wal 1340 ∈ wcel 2135 Vcvv 2721 ⦋csb 3040 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-sbc 2947 df-csb 3041 |
This theorem is referenced by: dfmpo 6182 |
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