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Mirrors > Home > MPE Home > Th. List > sbcng | Structured version Visualization version GIF version |
Description: Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Ref | Expression |
---|---|
sbcng | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3776 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥] ¬ 𝜑 ↔ [𝐴 / 𝑥] ¬ 𝜑)) | |
2 | dfsbcq2 3776 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | notbid 317 | . 2 ⊢ (𝑦 = 𝐴 → (¬ [𝑦 / 𝑥]𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) |
4 | sbn 2269 | . 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | |
5 | 1, 3, 4 | vtoclbg 3535 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1533 [wsb 2059 ∈ wcel 2098 [wsbc 3773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-sbc 3774 |
This theorem is referenced by: sbcn1 3829 sbcrext 3863 sbcnel12g 4413 sbcne12 4414 difopabOLD 5833 bnj23 34480 bnj110 34620 bnj1204 34774 sbcni 37715 rspcsbnea 41734 frege124d 43333 onfrALTlem5 44123 onfrALTlem5VD 44466 |
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