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Mirrors > Home > MPE Home > Th. List > Mathboxes > alrimii | Structured version Visualization version GIF version |
Description: A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
Ref | Expression |
---|---|
alrimii.1 | ⊢ Ⅎ𝑦𝜑 |
alrimii.2 | ⊢ (𝜑 → 𝜓) |
alrimii.3 | ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓) |
alrimii.4 | ⊢ Ⅎ𝑦𝜒 |
Ref | Expression |
---|---|
alrimii | ⊢ (𝜑 → ∀𝑥𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alrimii.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | alrimii.2 | . . . 4 ⊢ (𝜑 → 𝜓) | |
3 | alrimii.3 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓) | |
4 | 2, 3 | sylibr 235 | . . 3 ⊢ (𝜑 → [𝑦 / 𝑥]𝜒) |
5 | 1, 4 | alrimi 2203 | . 2 ⊢ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜒) |
6 | nfsbc1v 3789 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜒 | |
7 | alrimii.4 | . . 3 ⊢ Ⅎ𝑦𝜒 | |
8 | sbceq2a 3781 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜒 ↔ 𝜒)) | |
9 | 6, 7, 8 | cbvalv1 2352 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜒 ↔ ∀𝑥𝜒) |
10 | 5, 9 | sylib 219 | 1 ⊢ (𝜑 → ∀𝑥𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 Ⅎwnf 1775 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-sbc 3770 |
This theorem is referenced by: (None) |
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