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Mirrors > Home > MPE Home > Th. List > spim | Structured version Visualization version GIF version |
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2396 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) |
Ref | Expression |
---|---|
spim.1 | ⊢ Ⅎ𝑥𝜓 |
spim.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spim | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spim.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | ax6e 2392 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
3 | spim.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 2, 3 | eximii 1828 | . 2 ⊢ ∃𝑥(𝜑 → 𝜓) |
5 | 1, 4 | 19.36i 2223 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 Ⅎwnf 1775 |
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-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: spimv 2399 chvar 2404 cbv3 2406 |
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