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Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > spsbcdi | Structured version Visualization version GIF version |
Description: A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
Ref | Expression |
---|---|
spsbcdi.1 | ⊢ 𝐴 ∈ V |
spsbcdi.2 | ⊢ (𝜑 → ∀𝑥𝜒) |
spsbcdi.3 | ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) |
Ref | Expression |
---|---|
spsbcdi | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbcdi.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | spsbcdi.2 | . . 3 ⊢ (𝜑 → ∀𝑥𝜒) | |
4 | 2, 3 | spsbcd 3751 | . 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒) |
5 | spsbcdi.3 | . 2 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) | |
6 | 4, 5 | sylib 217 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 ∈ wcel 2106 Vcvv 3443 [wsbc 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-sbc 3738 |
This theorem is referenced by: (None) |
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