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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 3003 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 2971 | . 2 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → 𝐴 ∈ V) | |
2 | sbcng 3003 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | biimpd 144 | . 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 2148 Vcvv 2737 [wsbc 2962 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-sbc 2963 |
This theorem is referenced by: (None) |
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