Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcni | Structured version Visualization version GIF version |
Description: Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
Ref | Expression |
---|---|
sbcni.1 | ⊢ 𝐴 ∈ V |
sbcni.2 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbcni | ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcni.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | sbcng 3818 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑) |
4 | sbcni.2 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | |
5 | 3, 4 | xchbinx 336 | 1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2110 Vcvv 3494 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3772 |
This theorem is referenced by: (None) |
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