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Mirrors > Home > ILE Home > Th. List > sbcn1 | GIF version |
Description: Move negation in and out of class substitution. One direction of sbcng 2944 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Ref | Expression |
---|---|
sbcn1 | ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2912 | . 2 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → 𝐴 ∈ V) | |
2 | sbcng 2944 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | biimpd 143 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)) |
4 | 1, 3 | mpcom 36 | 1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1480 Vcvv 2681 [wsbc 2904 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-sbc 2905 |
This theorem is referenced by: (None) |
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