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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 3694 | . . 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 326 | 1 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∈ wcel 2107 Vcvv 3398 [wsbc 3652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-v 3400 df-sbc 3653 |
This theorem is referenced by: (None) |
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