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Mirrors > Home > MPE Home > Th. List > sbcn1 | Structured version Visualization version GIF version |
Description: Move negation in and out of class substitution. One direction of sbcng 3767 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 3727 | . 2 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → 𝐴 ∈ V) | |
2 | sbcng 3767 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | biimpd 228 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)) |
4 | 1, 3 | mpcom 38 | 1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 Vcvv 3431 [wsbc 3717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3433 df-sbc 3718 |
This theorem is referenced by: (None) |
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