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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 3775 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥] ¬ 𝜑 ↔ [𝐴 / 𝑥] ¬ 𝜑)) | |
2 | dfsbcq2 3775 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | notbid 320 | . 2 ⊢ (𝑦 = 𝐴 → (¬ [𝑦 / 𝑥]𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) |
4 | sbn 2287 | . 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | |
5 | 1, 3, 4 | vtoclbg 3569 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 [wsb 2069 ∈ wcel 2114 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3773 |
This theorem is referenced by: sbcn1 3824 sbcrext 3856 sbcnel12g 4363 sbcne12 4364 difopab 5702 bnj23 31988 bnj110 32130 bnj1204 32284 sbcni 35404 frege124d 40126 onfrALTlem5 40896 onfrALTlem5VD 41239 |
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